If you have the time and patience for it, you can do it. Mind that France is one of the most annoying Eurail/Interrail countries, with lots of mandatory reservations and rather inflexible booking, and infrequent connections.
To travel validly with TGV InOui, you need a seat reservation + a valid pass. You can check shortly before if there’s any availability at the ticket office or online (https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish → only for domestic journeys in France).
If not available, the TERs are a much slower alternative, usually not that frequent. But you can get somewhere, often via more interesting routes than the rather boring high-speed lines. Plenty of smaller towns and villages to discover that way, but you need time.
Early december on weekdays it’s probably going to be okay to find free places on most trains. (Fridays and Sundays might be a bit harder).
If you plan to travel around Christmas: forget it to hope on as you go. Reserve asap, as demand will be huge.
If you plan international TGV’s (e.g. to/From Germany), best reserve in advance or use the workarounds.
But mention your (rough) travel plans in order to help you advise for the workarounds for crossing borders. Community advice can save you a lot when crossing the Belgian, German or Swiss borders from/into France ;)