We consider the regularity of non-Maxwellian solutions to the stationary linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. Suppose that a solution has a bounded weighted $L^2$ norm in space and velocity with the weight of collision frequency, which is a typical functional space for existence results for boundary value problems (Guiraud, Esposito-Guo-Kim-Marra). We prove that this solution is Hölder continuous with order $\frac{1}{2}^-$ away from the boundary provided the incoming data have the same regularity and uniformly bounded by a fixed function in velocity with finite weighted $L^2$ norm with the weight of collision frequency.