In this paper, we are interested in a model of exact asymptotically flat charged hairy black holes in the background of a dilaton potential. We study the weak gravitational lensing in the spacetime of a hairy black hole in the Einstein-Maxwell theory with a nonminimally coupled dilaton and its nontrivial potential. In doing so, we use the optical geometry of the flat charged hairy black hole for some range of parameter γ. For this purpose, by using the Gauss-Bonnet theorem, we obtain the deflection angle of the photon in a spherically symmetric and asymptotically flat spacetime. Moreover, we also investigate the impact of the plasma medium on weak gravitational lensing by an asymptotically flat charged hairy black hole with a dilaton potential. Our analyses show the effect of the hair on the deflection angle in weak-field limits.