It's an interesting question! I hadn't thought about putting out food that had been removed from a large hoard before

However I don't think it's a good idea. I know it seems like a waste of food, but hamster food isn't massively expensive and it's part of their way of life, hoarding food and foraging every day. The problem with food that's been removed from a hoard is partly that it will have the hamster's scent, yes, but they will also know it's their stolen food! And it could get them confused and stressed. Aside from that I don't think the hamster would take it, because they always seem to know if it's fresh or not. I know even if the bowl is full of food, if you leave it a day or two they still won't take the food, but if you put new food out they will. Fussy little things!
I think it's best to leave the hoard alone, unless it's pee'd on, until it's time to do a proper clean out at that end. And then, as Socks Mum says, prune the bottom half out, which is the oldest food and throw it. As their hoards can get quite large. But their hoard is their security - they become very anxious without a hoard. But always leave some of the hoard behind. If it's been pee soaked then try and save a few pieces of hoard that smell familiar and then add a handful of new food to the hoard. They really need a hoard!
I would be the same if someone cleared out my food cupboard and there was nothing in the house. It would make me buy twice as much stuff and hide it in different places! Indeed I have done that with chocolate before

OH kept finding the hiding places.