In my experience there is no effective way to 'warm things up', although you can certainly mitigate or attenuate specific qualities to make things sound less harsh/brittle/cold.
Warmth comes from so many things, and while it does have a tonal aspect that we know when we hear, it is imho primarily an emotional quality that lives within the music itself and that expresses itself through the frequencies and transients.
So now you look at the color of the individual elements and their harmonics, and even more so in the relationship of the elements to one another. The way the sounds layer up, the relationships you create between them with the faders, pan knobs, eq and fx... all of this, combined with the sounds themselves, the voicing of the chords, the relationship of the melody to the tonic, the timing and the energy of the performances, these things are huge.
Space is also a gigantic factor in the equation. How much there is, what that space feels like, what shape the ambience(s) has/have. Finally, in addition to all the above I'd say a great deal of how warm a track presents itself to be comes down to the bass tone and the relationship of the bass to the rest of the mix. The mids need to be in reasonable proportion, and the texture of the high freqs is paramount; get that last part wrong and you blow the whole affair.
Your Speck is not remotely 'sterile', sterile implies a lack of something to the degree that this lack is detrimental or oppositional to the cause, and in no way is the neutrality of the Speck a detriment to achieving warmth.
I would say that you should be able to get something in the ballpark of 'warm' by pushing all of your faders up with no processing on the tracks and achieve a rough but meaningful balance of sounds. If you can't do that, then you're already embarking on a fool's errand by trying to throw gear at the problem when the problem is more fundamental and lays with either a) your ears, or b) the music, c) the recorded tracks, or d) all of the above.
I also think warmth is overrated; more accurately, I think it is pointless to chase warmth, or any sound or ideal, because at the end of the day you will hear the way you hear and that will determine the results you create. So my advice, fwiw, is to take what your ears give you and go deeper into it, whatever it may be, and seek out the aspects that make it unique, compelling, engaging, magnetic, and memorable. If warmth happens to show up the party then that's great, but it is not in any way necessary or even helpful if your goal is to make art that resonates.
Or, to sum up all of that rambling into one concise sentence: Do not waste time trying to manufacture "what should be"; instead, give everything you have into magnifying and intensifying "what is there."
Gregory Scott - ubk