So yes... you will need a work for hire agreement signed. You should always agree and sign it before the studio work begins.
How the details work is between you two.
You can pay him a straight up session fee for the song. He sings it and gets his pay. Normally between $50 to $150 depending on the amount of work. He signs away ALL his rights to get that paycheck. What you do with it then is all yours. He has no future interest in the song and no rights to it. The agreement should specify that clearly. He's selling you his rights...all of them.
OR....
You can agree to give him a percentage of future potential income.....if any. I have worked this sort of agreement with a few singers. We drew up an agreement and signed it, that I had complete control and all rights to exploit the song. They had no rights to use the song. they received no money for the session work BUT they were entitled to a fixed percentage of income from the song as a session singer, with no claim to ownership or writing credits. If the song is published, used in film & TV or recorded by an artist, they will make far more than they would have made if they had simply been paid as a session singer.
Some singers will go for that and some will not. If they are working session singers, they will likely want cash up front and not worry with the residuals. A friend who is local, sings well, and is not singing for a living is more likely to take the percentage offer.
That is how the business operates. Nothing stingy about it. You could send your backing track with a rough vocal and copy of the lyrics to any demo studio in Nashville Tn, and have a pro demo singer sing it for under $100 and they would sign the agreement stating that they relinquish any and all claims to the music. That's a fact! Session singers work cheap, and you can get one all day long under $100 for a song.... They figure about 1 hr to lay it down and get it right.
If he starts to sound like he's changing his mind and getting greedy.....like wanting up front and a percentage..... tell him nicely that you thank him for the time he did singing, but that is not going to happen.....especially if he's already been paid.
Of course... it's totally up to you....and if he's a really good singer and helps with writing too.... it may be worth the percentage.