What most naturally follows?

How to find a home for a cat
Verify that the cat is actually a stray.
Of course it might be you are followed home by a stray and the cat really needs a home. However, you may have been followed home by a cat that is simply lost and needs to find its way home.
If is important to determine which is the case before you move forward. Take the cat to the local vet to be scanned for a microchip.