Sleuth About Town
Leland, a contact is a townsperson that will never clam up on you. The physical evidence townies (those who help you link the pieces of evidence you have with your suspects) are particularly useful as you won't have to go through the list of suspects will them; instead, they will just tell you who the piece of evidence belongs to.
In each city you can get a maximum of one physical evidence townie and one witness evidence one (the ones that you can ask if they know anything useful about the case).
To get a new contact you will have to solve favours. Eventually, a townie will stop awarding you equipment and will simply become your contact. Forever. The next time will do a favour for him, however, he will be rewarding you with equipment again. To get a second contact is notoriously time consuming but you will eventually get it. I believe having a lot of experience usually helps.
Since we are on this matter, to get a favour you need to solve 4 cases. If you solve your favour, that counts as a case - you will get your next favour when you solve 3 more regular cases. If you travel, however, you will lose the favour you are due for and will have to solve 4 cases again. Naturally, the same goes for quitting a favour or making a false accusation.
Hope this helps, best of luck!