I hate to be like this, but the word "warm" also means a lot of things to a lot of people. :) Seriously though I'm not trying to be difficult and I want to help. It's not your fault, it's just that a lot of the the words we've all been taught to use to describe audio characteristics aren't really all that well-defined when we can't be there to hear specifically what the amp is doing.
However, I'll say that yes, I personally find that compressors almost always go a very long way with clean tones. A good compressor can remove or smooth out the harshness of the ear-piercing "bang" that happens when you hit notes hard by taking the loudest parts of your playing and quieting them down, and they can also increase the sustain of notes you want to ring out by making the quieter parts of your playing louder.
Another fix in a pinch is that you can simply turn the treble down and the gain and/or volume up. Turning the Treble down can go a long way towards smoothing out harshness, but our brains tend to prefer brighter tones in general so when you immediately turn the treble down, it might sound "worse" to you in that moment, so I'd encourage you to wait after turning the treble down before judging. Play around with that darker sound for a good minute or so before making your decision.
So yeah I think you're on the right track. First, mess around with your amp's EQ to dial the amp to be a bit darker overall and see if that works. But if it doesn't, then yeah definitely try a compressor.