The staffless combo you have is an old one that's used by some people because it's consistent. However, it's relatively difficult (236C dash 5B is harder than the equivalents in other combos) and there are combos that deal better damage/get more meter.
The staff combo you have is very commonly used because it gives you an extra bit of time to hitconfirm into the corner properly (you do 5B[m] 2C[m] 6D for corner combos) and it does more damage than the standard combo if you hit your opponent in certain places in the air. However, you should, for the most part, do this:
With staff: 5B[m] 5C[m] 3C[m] Tsubame
You can do this off of any starter and it does from 2k to 2.8k, depending on how you start it.
Staffless: 5B 6B 5C 2C 6C(1) Chuun 2C 5C j.B dj.BC Hatsu Chuun
You can do this off of everything except for 2B. 2B is... complicated as Litchi has a conditional gatling with 2B and 5B.
Litchi's staff confirms aren't too difficult. Midscreen they're rather easy as it's mostly based on where you're positioned on the screen. The CH confirms are a different ball game, however. Staffless confirms aren't too hard either, and change based on whether or not the opponent is crouching, where you are, and where the staff is. The tough confirms are the staffless-to-staff transitions, as they depend on a lot of factors.
I made a guide for the corner combo in the past, so I'll just paste that.
Staff corner combo:
Note: 5B[m] is unnecessary in all of these combos, but it is usually a better idea to use it whenever possible. You will usually skip it on certain air hits or when you're starting from staffless and you've used up too many gatlings to get 5B[m]. 2C[m] is unnecessary if you hit opponents in certain places in the air; you're just going to have to feel this out, but if your opponent is on the ground, you must always use 2C[m] or 6D won't hit them properly.
You have to do this pretty damn close to the corner. The staff has to go down maybe a couple of character lengths from the corner. If 6D wallbounds but the staff does not land close enough to the wall for you to combo onto it, use the "really bad starter" combo, regardless of what you started with.
GOOD STARTER
5B[m] 2C[m] 6D TK Hatsu Haku Chuun Haku Hatsu Riichi A Ippatsu A Ippatsu B 5D slight hold Chuun staff2 6D 5B 6C(1) 4kote j.B dj.BCD falling j.C 66A
MEDIUM STARTER
5B[m] 2C[m] 6D Hatsu Riichi A Ippatsu A Ippatsu B 5D slight hold Chuun staff2 6D 5B 6C(1) 4kote j.B dj.BCD falling j.C 66A
REALLY BAD STARTER
5B[m] 2C[m] 6D Hatsu Haku Chuun 5B 6C(1) 4Kote j.B j.BCD falling j.C 66A
4D STARTER (I came up with this one because no better and consistent alternatives existed when CSEX came out. There are probably better combos out there now.)
4D 5B 6C(1) 4kote j.B dj.BCD falling j.C 66A 5C[m] Tsubame
A good starter is staff2, 5B[m], 2C[m], air hit 6B[m], j.C[M], 5B, 6B, 5C, 2C, or any RC'd slow normal (like 2B[m] or 6D). 5B[m] 2C[m] 6D(1) RC 2B[m] 6B[m] 6D counts as a good starter, too.
A really bad starter is 2B (This is barely even possible but it's doable), more than one A or one where you're transitioning from staffless to staff and you do a lot of stuff before you get to the 2C[m].
A medium starter is anything else, including starters like CH 6C[m] IAD j.B[m], which, while starting on a "good" move, prorate too far for you to do the full combo.
Note that there is a special combo you can do off of 6B[m] and yet another that you can do with an RC, but these are not that important as they only yield a little bit of extra damage and they're a lot harder. You will also inevitably learn that it is possible to do the good starter combo off of 5A, 2A and throws but that it is very difficult, so I tend to just do the medium starter for those cases.
If you do the wrong combo, the combo will drop at 5D slight hold Chuun. This is a really, really bad time for a combo to drop as Chuun is both slow and unsafe on block; if you are not sure what combo to do, it is better to guess conservatively.
As for how to do this combo, the inputs are a little bit oblique and a lot of weird things can happen, so I'll break it down. Note that the medium/bad starter variants are just shorter versions of the long combo; if you can do "good starter", you can do the other two just fine.
5B[m] 2C[m] 6D
Super easy. Just input this.
TK Hatsu Haku Chuun
This is also pretty easy. You input this rather quickly. Because they're rekkas, the timing is pretty loose.
Haku Hatsu Riichi A Ippatsu A Ippatsu B
Things start to get real here. There is quite a bit of a delay between the Chuun from the previous section and Haku. However, delay too much and they'll bounce off the ground and the combo will drop. You want to input the Riichi A while you're hitting your opponent with Hatsu; you basically have to do this instantaneously. The input'll look like this: 236B63214A. After that, input Ippatsu B as soon as the game will let you. If you wait too long, the combo won't drop (unless you wait a reeeeeeally long time) but you'll start comboing them out of the corner.
5D slight hold Chuun staff2 6D
Press and hold 5D, and, while still holding 5D, begin to input the 236C for Chuun. Then, press C and release D immediately afterwards, wait a few frames, and then input 6D. You can actually hold 6 and mash D if you want, as D, by itself, does nothing when the staff is in the air. This part can be tricky, but once you get a feel for it, it's actually one of the easier parts of the combo.
5B 6C(1) 4kote j.B dj.BCD
This is another common thing in Litchi combos. The only difficult part of this is that 5B 6C(1) has to be done very, very fast. 5B has very little hitstun.
falling j.C 66A
You are most likely to drop the combo here. The falling j.C has to be delayed quite a bit for this to connect. After that, input a sliding 6A where you buffer the dash during your landing frames. If you input the dash on the ground, this will drop. If you don't dash, this will drop against pretty much everyone except Tager.
People do not provide staffless corner combos because you basically have to improvise. I guess a brief summary would look like this.
- Staff nowhere in sight, or set further than half of the screen away?
5B 6B 5C 2C 3C Haku Chuun Hatsu Haku Chuun 5B 6C(1) 4Kote j.B dj.BC Hatsu Chuun
- Staff closer than half of the screen away but not in the corner?*
Do whatever and then Haku Chuun Hatsu Haku Chuun 5B 6C(1) 4Kote j.B j.BCD falling j.C 66A.
Example: 5B 6B 5C 2C 3C Haku Chuun Hatsu Haku Chuun 5B 6C(1) 4Kote j.B j.BCD falling j.C 66A.
- Staff in the area near the corner but not as far in the corner as possible?
Do as few moves as possible and then an appropriate staff combo from Haku Hatsu. You will probably need to do the medium starter combo unless you hitconfirm this very well. You may even need to do the bad starter combo.
Example: 5B 6B Haku Hatsu Riichi A Ippatsu A Ippatsu B 5D slight hold Chuun staff2 6D 5B 6C(1) 4kote j.B dj.BCD falling j.C 66A
- Staff pushed as far into the corner as possible?
Do as few moves as possible and then this: Haku Hatsu Riichi A Ippatsu A 244 j.B Hatsu Haku Chuun 5B 6C(1) 4Kote j.B j.BCD falling j.C 66A. This won't work off of a bad starter.
Example: 5C 2C Haku Hatsu Riichi A Ippatsu A delay 244 j.B Hatsu Haku Chuun 5B 6C(1) 4Kote j.B j.BCD falling j.C 66A
- Staff in the air?
Worst situation you can be in, really. Takes a lot of practice. You NEED to land a 2C[m] (a 4D will also work against crouching) to do the proper corner combo; a staffless 2C will not work. The amount of moves you will need to do to get a 2C[m] depends on how far the staff is away from you. But the idea is to do *insert staffless moves here* 2C[m] 6D corner combo. The other thing to be wary of is that staff2 does not have a lot of hitstun and can cause your combo to drop.
Example: 6A staff2 6B 2C[m] 6D Hatsu Riichi A Ippatsu A Ippatsu B 5D slight hold Chuun staff2 6D 5B 6C(1) 4kote j.B dj.BCD falling j.C 66A
...Good luck with all that. Litchi's corner stuff isn't that hard in general, outside of the improvising.
*If the staff is further out, j.C 66A will drop. You can do this on purpose for oki if you want but it's not that great and most people won't fall for it more than once.