There's a rental car shortage this summer.
"Travelers report sky-high prices and sold-out dates even in non-beach destinations like Kansas City, Houston and Memphis. Even travelers with reservations complain that they now sometimes show up and, with no cars on the lot, must wait for a car to be returned and cleaned before they can drive off."
That's happened to me before with business travel. I've asked to have a dirty car in order to get to a work appointment on time.
Most flights into JAC are connections. If your incoming flight to SLC, MSP, etc is late, you'll have to wait for another connecting flight. That could put you 2 to 4 hours "late" - IF there's room for you on a connecting flight.
It's not a good idea to drive through the parks at night - for the safety of the animals in the parks and the occupants of your car.
Something to consider. To be safe, I would spend the first night in Jackson or GTNP before driving on to Yellowstone. Enjoy landing at one of the most beautiful airports anywhere and enjoy the first hours of your vacation in a beautiful surrounding. If you have to, you could cab or Uber to your hotel the first night and return to pick up your rental car when it's ready.
Also, Hertz has lost their lease at the JAC airport and Avis is moving back in. If you rent from an offsite vendor, it's not just outside the airport but a 20 minute (without traffic) drive away. That 40 minute round trip will add about an hour to your departure. More if you include the wait in line. At JAC, one person can wait for luggage while the other is in line for a rental car. At an offsite location, all on the shuttle van hit the lines at the same time.
It's about 1.5 hours from Jackson to the south gate and then 1 to 5 hours to YNP lodging depending on the lodging).
That IS happening nationwide right now.