The second step is to place a piece of paper over this and mark the corners of the hex, then maybe draw up the hex, and mark where streets will need to cross the borders.
Then, what's inside is fair game. It's only the edges that need to line up with matching streets.
In a few cases, I've drawn streets that form dead ends as they cross into a new hex. Where possible I've limited these dead ends to the street two in from the corner (moving counter-clockwise around from each corner, or four streets in from a corner if you're going clockwise), this ensures a dead end on one hex doesn't connect to a dead end on a second hex, and thus leave us with that chance of an orphan street not connected to any other roadways.