I'm not sure I agree about the punish bit. Halo is too fast for a ten second cool down.
It already does augment jumping, you can thrust in mid air.
A boost in height might as well just be an increase in jump height. If you're going to add double jump it should also be able to change your direction.
You can change directions by thrusting normally while airborne. One is purely horizontal movement while the other is purely vertical movement. Like running is horizontal and jumping is vertical, thrusting is horizontal while thrust-jumping is vertical. Augmenting classic halo abilities.
I posted this earlier:
Thruster and Map Design
Thruster should be modified so that tapping Jump while airborne gives a slight upward boost.
Thruster would be used to clear three main obstacles.
1. Crouch-Thrusting would be used to quickly slide under a crouch height barrier.
2. Jumping forward then Thrusting forward would be used to clear a wide horizontal gap.
3. Jumping upward then Thrust-Jumping upward would be used to clear a high vertical gap.
When a map has multiple obstacles, that each lead to different a path, players must choose which obstacle they wish to clear with their single-use-before-recharge Thruster, in turn choosing which path they wish to take. Of course, the majority of a map's jumps could be accomplished without Thruster but players who use their advanced movement ability to quickly move through previously untraversable areas can gain an advantage over opponents who choose to take the scenic route.
Using Thruster to quickly move through a previously untraversable area allows players to gain a positional advantage over their opponents but leaves their Thruster recharging while they are most likely to encounter an enemy. A player who saves his Thruster for use in battle can thrust sideways like a super strafe, thrust forward to melee an enemy, or thrust backward to avoid damage from an exploding grenade.
Giving players the ability to use Thruster in an offensive manner, in a defensive manner, or for simply traversing the map allows for a wide range of possible outcomes in gameplay without disempowering the player or adding randomness to the game.
To;Dr
There are three states a player can be in: crouched, standing, or jumping. Thruster already augments all three of those states but only adds horizontal movement to them. Adding a "Thrust-Jump" that moves the player upward adds a vertical movement element to Thruster.