This paper deals with the problem of sampled-data gain-scheduling control design for affine quasi-LPV systems. As the control implementation is based on sample-data and its update occurs only at specific sampling instants, the state-dependent scheduling functions of the controller are piecewise continuous. This characteristic causes a mismatch between the system model and the controller parameters, namely asynchronous scheduling functions of the controller with respect to the plant during the inter-sampling behavior. To cope with this phenomenon, a polytopic description of the inter-sampling is constructed based on bounding assumptions on the scheduling functions and their time derivatives. Then, regarding a time-delay approach, a constructive and numerically implementable LMI-based synthesis condition is derived, and a convex procedure is proposed for enlarging the estimate of the region of attraction of the origin of the closed-loop system. The time-derivative bounds of the scheduling functions are explicitly accounted into the local analysis to ensure the method's implementation. A numerical example is provided to illustrate the proposed methodology.