1) You shouldn't force 320x240 to 1280x1024. They are not the same aspect ratio, so you're likely stretching the screen.

2) You should stick to integer scale factors when forcing resolution. If you wanted to output at 1280x1024 for example, you would set rendering resolution to 320x256, since you can multiple both by a whole number to get 1280x1024 (320 *4 / 256 * 4). You get more graphic glitches when doing non-integer scaling.

3) But you should generally speaking avoid 5:4 aspect ratio (which is what 1280x1024 is), a lot of games render it wrong.

4) dgVoodoo, and last time I tried Peixoto's patches, shouldn't be used for forcing resolution for Jedi Knight or Outlaws because of the way that engine works. Here's an illustration of that: https://www.youtube.com/watch?v=h4NbVc80D-A

5) Speaking of Peixoto's patches, you might want to try those, they're tailored more per game, but I couldn't get Jedi Knight working with them.

6) dgVoodoo doesn't have any options other than changing texture filtering and disabling Bilinear blit stretch (it's disabled by default), so if that doesn't work you're out of luck.

7) Because of 4) To render this game in a higher resolution you should be using JkGfxMod:
https://github.com/jdmclark/jkgfxmod
This will actually render the game in a higher resolution, and scale the UI by zooming in on it. It has issues with the minimap, but everything else looks so much better. Here's a video showing a bit how it looks (watch at 4K when that's processed):
https://www.youtube.com/watch?v=b6XnKDg5BQo
That's 1280x960, with FOV mod set to 95 and the mipmap patch.

JkGfxMod has the same UI problem, it's linear filtered by default, but you can change line 418:
gl::set_texture_mag_filter(gl::texture_bind_target::texture_2d, gl::mag_filter::linear);
to
gl::set_texture_mag_filter(gl::texture_bind_target::texture_2d, gl::mag_filter::nearest);
in renderer/opengl_state.cpp and recompile it. Or if you trust me, just use the attached renderer.dll.

Thank you ZellSF!

Good observation about whole values. I've gone for 1600x1200 as that's 240x5. I don't have so critical an eye to notice too many glitches, but I will absolutely check out JkGfxMod. I went straight to dgV as I've used it so effectively in the past. But JkGfxMod looks like it will give me exactly what I need.

EDIT: It works beautifully! Thanks so much!

Doesn't dgVoodoo override the game's internal resolution for direct3d/glide programs? Does changing the in-game resolution do anything?

In-game resolution and forced resolution gives different results. Forced resolution gives more graphic glitches, but allows you to run higher resolutions than supported or it can be used for UI scaling, compare OP's 1280x1024 screenshots to this 1280x960 in-game resolution screenshot:

Of course as mentioned, Jedi Knight is one of the games where resolution forcing gives so severe graphic glitches (jittery 3D in this case) that it shouldn't be used. JkGfxMod doesn't force the resolution, it just zooms the UI parts of the game.