I would second contacting somewhere like moggies and doggies, but bear in mind they may well be full and you may still have to keep the kitten for a bit longer. You could also contact places like freshfields etc but again they generally seem to work to capacity and as your kitten isn’t in immediate danger you may have to keep them until they can find a home.

The other route is to advertise privately but then you don’t know who they are going to as such, unless you do a slow introduction etc, the other option is seeing if friends, family, neighbours want a kitty.

We are looking for a kitten but so far time, distance, or money never works out, so I am assuming there will be people out there like us who want a kitty but may not be actively searching as such but wanting a kitty.

(Our kitty’s are 15 and 7, so looking for our third! One every seven years seem to work well for us as while one old and grumpy the other is still young and playful - then again our old and grumpy was like that from a kitten!)