I can think of several things you can do with the glut of UCG ... how about giving some to another gardener who might not know about the benefits of UCG .... or how about sharing with an older housebound acquaintance who may not be able to pick up the wonderful freebies?

Other than that, when I have more than I can distribute around the garden, I just toss them on top of my compost pile. [especially useful to de-stink a bucket of kitchen scraps and keep the varmits away from my pile] All br/gr balance aside, my cold/no-turn pile somehow turns out some pretty good compost at the end of a year, even with a slap-dash approach.

"Should I spread the coffee grounds directly on the garden and lawn or mix it into my tumbler full of composting material?"

Yes. Just not too thickly because they can form a crust and repel water.

" Is the season a factor? "

A little. If you spread them on the lawn when it's too cold, they'll still be visible in the spring. So you have to stop at some point or you'll get them too thick (see comment above).

I put a layer about 2-3 inches thick around some perennials and let them form a firm crust. Then broke up the crust into 1-2 inch chunks and this served as a nice mulch. The chunks did not crust again.