Got a thumb strap that's new-ish, and while I know years of use will make it more comfy.... Any way to speed up the process?

Play hard for 4 hours a day. About day 3 it should start loosening up.

Seriously, I bought some small squares of soft leather at an arts & crafts shop and cut my own. Surprisingly, there has been minimal stretching and no break-in time.

Mink Oil?
Works on old, hard-as-iron work boots and dried-out belts, gloves, chaps, and so forth.
The hard part is catching and squeezing the oil outta them minks.
Nose grease also works. Not kidding. Put those blackheads to work.

JB

JB,
You got a laugh out of me with the mink squezzin' line. Good on you.:)
Thanks.

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Lexol.
That will make any leather soft, right down noodley soft, scarily noodley soft, but it will, once again dry, be the nicest leather forever.

I have used Lexol on many bridles and some saddles, some new.
One that it scared me how dark and too soft it got, but cleared up again and the leather is now, 40 years later, still in excellent shape, if darker by now.

Still, as someone mentioned, now knowing what kind of leather you have there today, the manufacturer would be your best bet for directions on how to care for it.
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These are not my words - I never heard about Lexol before - but of horse people, as you see. But apparently it works!!