I am told there is a way to do this- but how?

is the easiest way.

It costs 150K in the millionaire center, and gives you 1 to 3 spaces by box.

Thanks for the quick response! That seems to be a little out of my price range.

You can also get additional spaces through random events.

And the Cash Shop item, Multi-Compartment Keepsake Box, gives 10+ spaces.

You can get some just randomly with your active pet, all mine were achieved that way.

I'm pretty sure the random event gives extra spaces to all your pets, not just your active pet. :)

, I was thinking more of the Oracle, but you are right about the random event.

I've gotten some free spaces from random events.

The pet zapper V2 (on Atebus) can also give treasure spots (among lots of other things), when I was zapping one of my pets the most frequent outcome for me tended to be a sex change, but I'd get a treasure slot every once in a while. If you use the zapper, beware - it can destroy inventory items which are not locked, and it can also turn your pet's minion into slime.