If you are an outstanding candidate I think the coach should be able to realize directly that you deserve a shot - this does not mean he will refer you. The ideal scenario is that on top of the case coaching he will teach you how to network and maybe even direct you to reach out to some of his connections in the offices you are aiming to apply to.
This is something I personally pro-actively do with my top mentees (middle east focused network). My top mentees know and understand that I would help direct them towards individuals from my network that would be willing to hear them out, but they understand that me telling them to reach out to X or Y individuals does not guarantee a referral in any way, they would still need to send a personalized message, wait for a reply, get on a call, prove why they deserve the referral, and then repeat the process with the other contacts in the same and different firms.
Any coach could simply reach out to his network and get the referral based on recommendation along but this would harm his/her credibility in the longer run.
For me, this is an important part of the application process and you should not be handed a referral on a silver platter just because you bought a coaching session or package (this is a recruiter's job, not a coach), this is something your coach might be willing to help with but it would not be the norm.