Second on Full Service Storage. Best in town. They have two locations, one near the airport which would be ideal for you. It's a great operation. Indoor, secured (gated as well as locked warehouse inside the gate), climate controlled, immaculately clean, and staff on site all day. And they can make any repairs you might want done while it's there, have it washed, etc. Only the staff is allowed inside, you unhook outside and they pull it in for you with a tug, and out when you return. So no worry about other people damaging your unit backing theirs in like at a normal storage place. I kept mine there for the first winter I owned it. It's huge, and has a lot of high end toys stored.
Also, if you arrive into SLC after 5 pm or when they are closed, rather than get a hotel, you can call in advance and they will pull your rig outside the warehouse, and give you the gate code so you can get in and retrieve at anytime day or night. They'll park it behind the building so that nobody driving by will even know it's out there.
They should accommodate you as they have a monthly option which is $9/foot per month for the trailer and a flat $115/month for the pickup. Much cheaper than a campground without the worry about security, weather damage, etc. Should be no problem during summer since many people get their toys out for the season and only store in the winter. They fill up in the winter.
https://www.fullservicestorage.com/