TBs only have to be tattooed if they're entered in a race or train at an actual official racetrack. Lots of breeding and training facilities have training tracks, it's likely Ginger trained on one of those in her youth.
I'm so jealous you moved to such a nice facility to ride in, with such nice trails as well as a track! I wish I had a track to ride on after training hours, it's so great for fitness. We used to have one nearby, we had to trailer to it, but it's closed now, unfortunately. It was so nice to trot or gallop, knowing exactly how much distance you're going, plus the footing is smooth and they're not going to step in a hole or trip on a rock and injure themselves.
You should definitely ride Ginger on the track! Since Ginger's an older girl but as long as she stays sound, take her around the track once or twice a week after hours, at a steady, balanced but forward trot and keep track of your time, don't make her exert herself so much that she sweats excessively or comes back breathing heavily. Keep yourself balanced but off her back as much as possible.
You'll notice after a few weeks, she'll trot around still steadily, but not be sweating at all. And the time she takes will be slightly faster. At this point you can add a bit of slow canter every other furlong, trot transitions in between.
Do your normal schooling in between 1 or 2x week track rides, and you will be amazed at her fitness - not only at how stronger her muscles get, but how enthusiastic she becomes about working with you on her arena work, and how much easier it becomes for her to learn new things, since she will find them easier physically.
I guess you can tell, I'm a big fan of you and Ginger!