Networking Call Follow ups

I've had some networking calls recently with some great analysts. Most have them have told me something along the lines of "don't hesitate to reach out if you have any additional questions." 

Since many of the places I'm looking at don't open up until July, I want to keep in touch with these people, but I'm not sure what an appropriate follow up conversation starter/question is. They answered all my questions on the call so would it be appropriate to email or call with questions about the application/interview process once its June? What else can I follow up about?

I'd appreciate any input/advice on this