The sequential regularization method (SRM) is a dynamic iterative method for the numerical solution of higher-index differential-algebraic equations (DAEs). The SRM has the advantage of being based on a regularized problem that is less stiff than those produced by standard regularization methods. Consequently, nonstiff integrators may be used, making the SRM a competitive alternative to popular integrators. In past work, the number of SRM iterations was taken to be roughly equal to the order of the numerical method used in each dynamic iteration. In this paper, we propose a predicted SRM (PSRM) that reduces the number of iterations in each dynamic iteration to one. We give a new error analysis for explicit Runge--Kutta methods applied to linear index-2 Hessenberg DAEs with or without singularities. We also give numerical examples to confirm the predicted convergence rates. For the PSRM, extrapolation formulas and methods based on the differential part of the DAEs serve as a predictor, and the SRM iteration serves as a corrector. Implementation of higher-order schemes for the PSRM makes use of continuous extensions of Runge--Kutta methods. In particular, we give a prediction scheme for the algebraic variable at intermediate stage points that suppresses order reduction in the differential variable near a singularity. Moreover, the SRM/PSRM provides new insight into operator splitting and fast convergence rates for waveform relaxation.