I don't know about NYC specifically, but often times bars host small shows that are basically free, you're just expected buy a drink (you probably can even skip that part if you really don't wanna spend money at all). I've seen some amazing musicians that way and to me, small shows like that are way more special than larger shows.
Check out the socials of bars near you and you should find something interesting, maybe even a new small artist that you will love!