For the Hyundai Tucson, can the spare be stored under the rear cargo floor when the foam insert is removed?

Yes. Please read below for further details.

For Hyundai Tucson Hybrid models, there is space beneath the rear cargo floor that serves as an excellent spot to store your Modern Spare tire. There are important details to keep in mind.  
In some model years, when the foam insert is removed, you might notice small bolts protruding from the floor of this storage area. To store the spare in this space and prevent these bolts from damaging it, you need to place some kind of protective material over the bolts before positioning the spare.  
The presence of these bolts differs by model year. If they are absent, no precautions are necessary.  
The Hyundai Tucson Hybrid also comes in a Plug-In Hybrid variant. These models may not offer this storage space, as it is utilized for the battery system and related components. For plug-in hybrid models, please opt for the kit option that includes a carrying case, specifically designed to store the spare kit in the main cargo or trunk space.