Yes you can if your weather stays above 50 at night and has daylight temps in the 70 and 80's during the day.
You might want to start new plants from cuttings when the existing plant gets too unruly or starts getting disease problems.
You can easily make cuttings from the sucker shoots. You will have to wait for the new plant to get established, but you can keep a plant going indefinitely with the right growing conditions.

Hi, welcome to the big online tomato patch,
I don't know of any place in Florida (your State) that you would want to keep a plant going. After a plant gives a good fruit load which I guess yours has yet to do, itâÂÂs usually pretty badly whipped and is highly susceptible to disease. That's the reason it is best to start from seed for me here in FL.

Gregory's suggestion is great if you want to 'keep it going', or grow in a Florida Room, greenhouse or other protection where you can moderate temperature swings, but if you are in South Florida the time to take the cuttings would be in early June; North-Central Florida early July, and grow out the sucker during the hot months inside the house.

I find cuttings a pain without a nice Florida room, etc., and seeds are so easy to start, and give a better no-disease guaranty, that although I thought I would be doing that, after this summer, I have reevaluated the options vs. work and prefer just to start from seed. You just get to watch the seedlings inside, which I actually enjoy since they are diminutive and manageable, and if it is an heirloom you can just save the seeds and replant fresh seeds ...

The one time I would take cuttings is for a short fall season (north Florida). It can give you a real running start. That was my plan this year, but the plants started to produce and I had disease concerns, so I left them be and they are already yielding fruit.

Hope that helps, since I basically think there is nowhere in Florida you can keep going with the same plant, except maybe the Keys where it never freezes. But the water consumption and near zero production make it slave labor to keep a plant going for minimal results, speaking from my experience this past summer.

PC