To attain good performance in an acoustic echo cancellation system, it is important to have a variable step size (VSS) algorithm as part of an adaptive filter. In this paper, we are concerned with the development of a VSS algorithm for a recently proposed subband affine projection (SAP) adaptive filter. Two popular VSS algorithms in the literature are the methods of delayed coefficients (DC) and variable regularization (VR). However, the merits and demerits of them are mutually exclusive. We propose a VSS algorithm that is a hybrid of both methods and combines their advantages. An extensive study of the new algorithm in different scenarios like the presence double-talk (DT) during the transient phase of the adaptive filter, DT during steady state, and varying DT power is conducted and reasoning is given to support the observed behavior. The importance of the method of VR as part of a VSS algorithm is emphasized.