The answer to your first question is "no": a finite amount of tape may not be enough. At the same time, the answer to your alternative question is "yes": once it finishes, a Turing machine has never used more than a finite amount of tape.
The fundamental issue is that in general, it's impossible to tell in advance how much, or whether it will finish in the first place.
Do Turing machines contain an infinite tape? I would say the answer to that question is "no".
Reason one: a Turing machine doesn't contain any tape, it uses tape. You can see this in its mathematical definition, which doesn't contain any tape or tape contents. The definition doesn't imply anything about how or when the tape it uses is supplied. It makes no difference to the Turing machine.
Reason two: a Turing machine never uses more than a finite amount of tape, in the sense that at any point during its computation, it can never have used more tape cells than it has made computational steps. However, at the same time, a Turing machine may use an infinite amount of tape, in the sense that there may be no bound on the amount of tape it will eventually use. Therefore, if you should start with a given, finite amount of tape for it to use, no matter how much you supply, it may not be enough: you may need to extend it on the fly, and you may need to do that infinitely often.
Therefore, only if we think of the tape as pre-supplied with the Turing machine (which we don't need to do), the only choice we have is to supply an infinite tape (which is clearly impossible).
When Turing invented his Turing machine, a computer was a woman making calculations on pieces of scrap paper. The nature of computation is such that there is no fixed limit on the amount of scrap paper (or tape) a computation may require. Nobody would ever say that a human computer is a woman (or man) containing an infinite amount of scrap paper, and therefore cannot really exist. Then why do I keep reading that Turing machines contain an infinite tape, and therefore cannot exist? It is wrong, or at least misleading: I've seen misunderstandings arise as a result. Therefore, I would prefer if courses on Turing machines would stop saying it.