It sounds like you are asking for a company who can do real estate sales as well as manage your property as a rental while its on the market correct?
There are a few brokerages who have agents that have a 'home tenders' company or that work with one. Then, there are agents who own a property management company or work with one.
In either case, there is a significant difference.
Home Tenders- A company who provides a rental based on a 'private sub-contractor agreement' for specific terms and conditions. They are not technically tenants who live in the home and they are there with a firm understanding that you are selling and they need to accommodate the process by making the home available to show and keep it in good appearance. Some companies even require that the sub-contractors have certain qualities of furnishings.
Property Management- A company who rents or leases the property and is held responsible for the terms and conditions of the lease by Idaho Code. They can find similar tenants but the contract is very different and may or may not be suitable to your situation.
If you'd like a referral to a couple of these companies, please let me know, I can connect you with a quality Real Estate Brokerage who knows the ins and outs of the Sale/Rent situation.
208-724-7602 or firstname.lastname@example.org