While they may influence how you handle future interviews, I'm not sure I'd see that as enough motivation for them to stay in contact with you. In wanting to build this relationship, what are you giving to them other than some of your time? Consider what are you offering, what do they want, and how well can you match these as while I can understand how such a relationship benefits you, how does it benefit them? That's the part I don't see at the moment and is the part that you have to resolve as that is the heart of your message in communicating them. For example, you could send an e-mail like:
I'd like to discuss your current challenges and how I may offer some
help in overcoming those challenges. Could we meet for lunch sometime
this week to discuss this in more detail? Thanks for listening to me.
Short and simple as what you want to have is a regular conversation that enables you to get a view of their operations and how you could fit in there. At the same time, by offering some assistance to their problems, you could work your way into having an offer when the school year is over as if you can prove useful on this part-time basis, imagine what you could do in a permanent role.