For a large class of functions f : Fq → E(Fq) to the group of points of an elliptic curve E/Fq, Farashahi et al. (Math. Comp. 2013) established that the map (u, v) → f (u) + f (v) is regular, in the sense that for a uniformly random choice of (u, v) ∈ F 2 q , the elliptic curve point f (u) + f (v) is close to uniformly distributed in E(Fq). This result has several applications in cryptography, mainly to the construction of elliptic curve-valued hash functions and to the " Elligator Squared " technique for representating uniform points on elliptic curves as close to uniform bitstrings. In this paper, we improve upon Farashahi et al.'s character sum estimates in two ways: we show that regularity can also be obtained for a function of the form (u, v) → f (u) + g(v) where g has a much smaller domain than Fq, and we prove that the functions f considered by Farashahi et al. also satisfy requisite bounds when restricted to large intervals inside Fq. These improved estimates can be used to obtain more efficient hash function constructions, as well as much shorter " Elligator Squared " bitstring representations.