Running 2nd sub panel from 1st sub panel
Can't run Romex (NM-B) outdoors, need UF-B. 

The "100 amp" or whatever on the subpanel is an absolute limit

OK so you live in Wyoming and the posted speed limit is 80 mph. The tire shop has tires rated for 85 mph, 112 mph, or 130 mph.  Would you want the 85 mph tires?  You'd be running them at redline all the time. 

And so it is with subpanels. 100A or whatever is the redline, as far as bus rating.  

The main breaker is irrelevant in a subpanel

As far as main breakers in the sub, that's only there to be a disconnect switch, so the amp rating on it doesn't matter.  

So ultimately the answer is "feel free to use a 200A-breakered, 225A-bussed subpanel fed off a 60A breaker".  

What would not be OK is powering a 100A-bussed panel off a 125A breaker. 

Is your panel too small?

If your basement panel is getting close to full, one option now is to upgrade it to a larger panel, and then upcycle the old panel to the garage.  

When planning breaker spaces, keep in mind NEC 2020 drops a bunch of new GFCI requirements that force even more (nearly all, at this point) breakers to be full-width breakers. Narrow half-breakers or 2-in-1 breakers (we call them double-stuff) can no longer be counted on.