One can construct t-structures on the bounded derived category of coherent sheaves on a smooth projective curve (or higher-dimensional variety) by tilting, see Bayer's notes, Prop. 3.6.1, and the corresponding hearts are not shifts of each other, because the tilt shifts some objects but not all of them.
Another way to obtain t-structures is to apply autoequivalences of the derived category to the standard heart, see Exercise 11 in Bayer's notes for the elliptic curve case.
Generally t-structures on bounded derived categories of coherent sheaves are studied in the context of the Bridegland stability conditions, and stability conditions for elliptic curves have been classified in Section 9 of Bridgeland's original paper.