Pseudospectral and collocation methods form a popular direct approach to solving continuous-time optimal control problems. Lifted Newton-type algorithms have been proposed as a computationally efficient way to implement online pseudospectral methods for nonlinear model predictive control (NMPC). The present paper extends this work based on a rank-one Jacobian update formula for the nonlinear system dynamics. In addition, we describe an algorithm implementation where this rank-one Jacobian update can be used directly to compute a low-rank update to the condensed Hessian, resulting in an overall quadratic computational complexity for each iteration. A preliminary C code implementation is shown to allow considerable numerical speedups for the optimal control case study of the nonlinear chain of masses.