I'm assuming you're using photoshop. If so, you should use the filter > noise> reduce noise option to get rid of JPEG artifacting.
Also, the way things like this work, you should be working in multiples of the original resolution, so if you want the 1080 image, you should be resizing it to 2048 x 1792(4x the source resolution) and the game itself would scale it back down to 1920x1440. That is only really beneficial though for doing 16:9 pan and scan. If someone is playing in 4:3, theres no benefit to that.
Also, here is my improved upscaling I've been working on in game.
The textures themselves are rendered out to 3x their normal size, and then ingame resized down to 1440x1080 (4:3 1080 resolution).