I'm still new at cake baking/frosting, so I haven't got all the kinks worked out.  

The last time I made IMBC the texture was lovely and smooth when it was first made.  But by the time I crumb-coated my cake, let it chill, applied the final coat of BC, and let it chill again before piping a border, the buttercream was starting to get very soft and a bit "spongy."  Would beating it again help to restore the smooth texture?  I don't think chilling it would be the answer, because I'd have to let it return to room temp again before beating it.

That is a common result when you let it sit a bit. I just use a fork or honestly, whatever is handy and give it a whip before spreading the next coat.

Because of the high butter content, if your kitchen is too hot, it can get a bit melty. if that happens, put your bowl inside another bowl of ice water and whip it. This will cool and firm up the BC without fully hardening it

Thanks.  My kitchen rarely gets hot, but I'll try whipping it if it gets a bit soft.  I just wasn't sure if that would harm it.

"spongy" is the word RL Beranbaum uses to describe the same phenomenon in her book, The Cake Bible.

so it's pretty common.

Yes, I know.  I use her recipe for IMBC.  But I don't think she mentions how to deal with it when it gets spongy.

Spongy when referring to a MBC makes me think it is too cold, but I know you didn't say that. So, I can't think of answer to give you other than yes, chill it again. If it gets too solid, put the bowl over a burner (stove) and melt it down a bit. Beat in the KA with the paddle attachment, and get it soft and silky again. 

I quite like it spongy.