It is wonderful that you took a rescue into your family. Thanks so much. Here are a few things I have heard people try over the years.
1. Keep him on a long leash (like 8-10 ft) tied to you or another responsible person when he is in the house. That way you can begin to identify clues when he is getting ready to mark and put him outside. Continue until the problem stops because he has learned what you want and wants to be off the leash.
If he was just recently neutered, it might be residual hormones behind this. So a few weeks might help.
I have heard people say a new dog in the house needs a room that is his. A place filled with his scent and no other animals. Helps he build confidence. I don;t know if this is true but when you are desperate, anything is worth trying.
There are underpants for bitches in heat (place for tail to come out). Some people have tried this when males piss inside as they absolutely hate it when they get in on themselves. The male dog wears them for a week after the last time it looks like he is going to raise his leg … Then if it reccurs, they go back on again (only when in the house obviously). Again I have not had this problem, but the panties did stay on an older bitch I had who was becoming incontinent. I thought she would bite them off but the ones I got were really well made.
Keep us posted on his progress. A loving basenji is worth a few cleanups!