In this paper we propose AbsTaylor, a simple and quick method for extracting inner polytopes, i.e., entirely feasible convex regions in which all points satisfy the constraints. The method performs an inner linearization of the nonlinear constraints by using a Taylor form. Unlike a previous proposal, the expansion point of the Taylor form is not limited to the bounds of the domains, thus producing, in general, a tighter approximation. For testing the approach, AbsTaylor was introduced as an upper bounding method in a state-of-the-art global branch & bound optimizer. Furthermore, we implemented a local search method which extracts feasible inner polytopes for iteratively finding better solutions inside them. In the studied instances, the new method finds in average four times more inner regions and significantly improves the optimizer performance.