The Red Lion hotel downtown off 600 South offers free long term parking for pilots. You just show up, let the front desk know you're there for the "park and fly" and they have you sign in on the clip board. Then the shuttle takes you to the airport. Tip the driver and give them a call when you return. I believe the shuttle goes every 20 mins starting at 4am, but you'd want to call to check.

Edit: I'm not sure about rates at slc, but I believe it's about $45 per month.