First of all, we usually don't say "tasse de café" but only "un café".
Secondly, I think the most natural way to say it would be: "Je voudrais un autre café (sil vous plaît)", but you could also use your other proposal, even if it sounds less natural to me ("Je voudrais encore un café").
But your remark is important: I can say phrases in a café I wouldn't use with friends (or only as a joke), whereas I can say in a café what I would use with friends.
So you'd better learn the polite and expanded way of speaking, gavarak.
« La même chose, svp ! » is a short sentence said in order to be quickly understood by the waiter and to have your dish quicker... that's all.
I would say un autre café, s'il vous plaît (the s'il vous plaît is pretty well mandatory). I think garavak's suggestions are more convoluted than most people would use. Like Isis34, if I were to expand my order to a full sentence, I would use je voudrais in preference to j'aimerais.