Prefiero tomar otro té caliente en la cafetería.

Prefiero is a direct verb meaning I prefer and is used to state a preference clearly and explicitly.
Me gusta más literally means I like (it) more and often sounds a bit softer or more descriptive.
Both can work, but prefiero + infinitive / noun is very common for straightforward preferences.

In Spain, tomar is widely used for consuming drinks (and sometimes food), especially in everyday speech: tomar un té / tomar un café.
Beber is also correct, but it can sound slightly more literal/neutral and is often used when emphasizing the act of drinking (or alcohol): beber agua, beber alcohol.

After prefiero, Spanish commonly uses:

• prefiero + infinitive to prefer an action: Prefiero tomar…
• prefiero + noun to prefer a thing: Prefiero otro té…

Here the speaker prefers the action of having/drinking another tea.

Yes. Prefiero otro té caliente en la cafetería is grammatical and means you prefer another hot tea (in that setting).
Adding tomar emphasizes the action/choice to have it: Prefiero tomar… can sound a bit more natural when ordering or deciding what to do.

Otro means another (one more, a different one).
It normally goes before the noun: otro té.
After the noun (té otro) would be wrong here.

Té (with accent) means tea.
Te (without accent) is a pronoun meaning you (object pronoun): Te veo = I see you.
So the accent distinguishes two different words.

Most descriptive adjectives in Spanish typically go after the noun: té caliente.
Putting the adjective before the noun can be stylistic and often changes nuance; un caliente té sounds unnatural in standard Spanish.

Caliente covers both warm and hot, depending on context. For drinks, it often means hot (as opposed to frío).
If you want “nicely warm” or “not too hot,” you might hear:

• templado = lukewarm/tepid
• calentito = pleasantly warm (more informal)

En la cafetería typically refers to a specific café that’s known in context (the one you’re in, or “the café” you both have in mind).
En una cafetería would mean in a (some) café, more general/indefinite.

• en la cafetería = you’ll drink it there, in the café.
• para la cafetería would mean for the café (purpose/destination), which doesn’t fit well here.
  If you mean “to go / take away,” Spain often uses para llevar: Un té para llevar.

Té is masculine: el té, un té, otro té.
You can tell from the article and adjectives/determiners used with it: otro (masculine) matches té.