We analyze the chemical and jump surface diffusion coefficients, D(c) and D(J), near a first-order phase transition at which two phases coexist and the surface coverage, θ, jumps between single-phase values θ(-)(*) and θ(+)(*). Contrary to other studies, we consider temperatures that are sufficiently subcritical. Using the local equilibrium approximation, we obtain approximate analytical formulas for the dependences of D(c) and D(J) on the coverage and system size, N, near such a transition. In the two-phase regime, when θ ranges between θ-* and θ+*, the diffusion coefficients behave as the sums of two hyperbolas, D(c) ≈ A-/N|θ-θ(-)(*)| + A+/N|θ-θ(+)(*)| and D(J) ≈ A(-)|θ-θ(+)(*)|/θ+A(+)|θ-θ(-)(*)|/θ. This behavior rapidly changes as the system goes from the two-phase regime to either of the single-phase regimes (when θ goes below θ(-)(*) or above θ(+)(*)). The crossover behavior of D(c)(θ) and D(J)(θ) between the two-phase and single-phase regimes is described by rather complex formulas involving the Lambert function. We consider a lattice-gas model on a triangular lattice to illustrate these general results, applying them to four specific examples of transitions exhibited by the model.