Kathryn Gumshoe the 7th
You do favors for the townies. But how long it takes you to get contacts is completely random. Sometimes it takes quite a while.
After you solve 4 cases (the favor counts, so it's only 3 more cases unless you don't solve the favor) you can do another favor. There is a link at the bottom of the questions you ask the townies that says something like "ask them if they need a favor."
Hope this helps.