My last-minute gifts are more like "nifty gifties". I have to admit I keep a closet where I stash things that are special but not overly personal, if that makes sense. The sort of thing that almost anyone would enjoy, but that weren't bought with a specific recipient in mind. You know, like a hostess gift. This is where I "go shopping" when I need a last minute gift.
So what's in the present closet right now... Lots of higher-end scented soaps and candles (I only use these when I know someone loves scents and isn't sensitive to them) and fancy boxed matches to pair with the candles; a variety of mugs stenciled with initials; a little gold Coach wristlet (not for anyone with a big phone); a variety of little handmade cloth bags from etsy, meant to hold tissues or makeup in a handbag; an assortment of my favorite little French paring knives and printed French linen dishtowels, for people whom I know do a little cooking. A holiday-themed soap-and-lotion set in lovely pump dispensers. Then there is always the last-minute bottle of wine or champagne, if I know my recipient enjoys that. I tend to buy these things when I see them and just tuck them away for when I need a gift.
I use the same closet to hide Christmas and birthday gifts for Mr. April and Little Mr. April so the rule is that after July or so, they're not permitted to open the door (they both have winter birthdays.)
It occurs to me that right now, I have nothing in that closet suitable for a child, so I'd better keep my eyes open for something to fill that gap.