Is it illegal to drop off a stray cat somewhere other than a pound?

Here in the you k it is illegal to cause unnecessary suffering and "dropping off " a stray cat in any strange place other than a rescue/sanctuary would  more than likely cause unnecessary suffering unless of course there was a kind stranger around to take it  over and deal with this situation and this is very rare.      So the best thing is to ring all around the local ( and perhaps national too) animal rescue places ie cats protection and ask if they can take it in. Another possible solution would be to take it to your local vet. For them to see if it is micro-chipped and possibly return to long lost owner. It does happen.   But please don't just dump it anywhere as that would be cruel. You really need cat rescue people to sort it out for you if you are not prepared to keep it yourself. So keep on trying them because they are pretty full at the moment.

You'd have to check the laws where you live but it often is and even if it's not it's still not something you should really do. Abandoning an animal other than somewhere which is designated for that purpose is usually illegal on top of the moral aspects. It isn't just the wrong thing to do for the cat, it has an effect on the whole area where the cat would be released and other people and animals that happen to live there. 
For a stray hanging around your house or if the cat was yours and you are unable to find a home for it you would take it to a shelter or rescue yourself. If they aren't open today then they probably will be tomorrow.  Or if a stray, you can call animal control or a rescue facility to come and remove the cat or put out a humane trap to do so if that's what's needed.
Make sure your daughter doesn't try to interact with the cat while it's hanging around if that is a concern. You can try to weigh down the lid on your garbage or keeping it in the garage except for trash day or something like that until the issue is resolved if that's a problem for you.
If there are a lot of cats in the area you might look into a trap/alter/release type program which  has a much better effect on controlling the population so there aren't just more and more cats in the feral population adding to the problems in your neighborhood. Good Luck.
www.cfainc.org
www.njaudubon.org

It's not illegal .. But it is immoral.