We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas--Its--Kitaev Riemann--Hilbert problem using the Deift--Zhou method of nonlinear steepest descent and its $\overline \partial$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that are slightly improved while less regular perturbations are admissible. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be twice differentiable, with bounded third derivative.