Without looking at the chart to see the exact disposition of things... if you need to fly through Class C before you get to the Class D, just call them. They will hand you off to tower before you actually get to Class D.
My airport (Palo Alto) isn't quite inside the Class C for San Jose, but close, with the added complication of another Class D (Moffett) in between. It all works smoothly 99% of the time (except like the other day, when the Norcal controller couldn't contact the Moffett controller, who was probably in the bathroom or something).