If I were a rookie, there are plenty of things a Chess coach could teach me: forks, pins, discovered attacks, sacrifices, and other ideas like protecting the king, controlling the center, avoiding pawn islands and preventing the opponent from having passed pawns, amongst other basic ideas.
But I am not a rookie anymore, and I'm not sure what a Chess coach could teach me other than openings, traps and end game calculations. Is the coach going to play against me and call all my blunders, which I could have seen on my own if I had paid more attention? Is he going to describe 10 move deep discovered attacks that I never would have seen on my own and, given the large number of variations, will probably never see again? Is he going to offer emotional support? Or is he going to teach me something that I can actually apply in all my games? If so, what?
I'm not a chess coach but I am a professional musician and teacher. I can tell you that I've often had private students who could play fine. To me it's a similar situation. As a coach/teacher in that situation my job is try to identify the "holes" in their mental approach to their music as well as figure out what their strengths are. It isn't really possible to make a "syllabus" until I figure out what needs to be addressed. My guess is that as a decent player a good coach would need a few sessions with you to really get inside your head to figure out how to plan for your progress. If you're looking for some sort of "boiler plate" approach then you don't understand the nature of true teaching.
He'd most likely want to analyse a few of your serious games. You would have tactic and endgame exercises / homework. He would go over how to evaluate a position and find moves that make sense in a given position. Give you history and theory behind the openings you play. Probably not play games with you, although analysis games can be a good learning tool -- more likely play out key positions with you like a themed middle game or endgame.
I agree with some others that you'd probably want to find a club and go to some tournaments.