We determine the smallest possible canonical height for a non-torsion point P of an elliptic curve E over a function field ℂ(t) of discriminant degree 12n with a 2-torsion point for n=1,2,3, and with a 3-torsion point for n=1,2. For each m=2,3, we parametrize the set of triples (E,P,T) of an elliptic curve E/ℚ with a rational point P and m-torsion point T that satisfy certain integrality conditions by an open subset of ℙ2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that for n=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.