  We develop an alternative time-dependent numerical renormalization group
(TDNRG) formalism for multiple quenches and implement it to study the response
of a quantum impurity system to a general pulse. Within this approach, we
reduce the contribution of the NRG approximation to numerical errors in the
time evolution of observables by a formulation that avoids the use of the
generalized overlap matrix elements in our previous multiple-quench TDNRG
formalism [Nghiem {\em et al.,} Phys. Rev. B {\bf 89}, 075118 (2014); Phys.
Rev. B {\bf 90}, 035129 (2014)]. We demonstrate that the formalism yields a
smaller cumulative error in the trace of the projected density matrix as a
function of time and a smaller discontinuity of local observables between
quenches than in our previous approach. Moreover, by increasing the switch-on
time, the time between the first and last quench of the discretized pulse, the
long-time limit of observables systematically converges to its expected value
in the final state, i.e., the more adiabatic the switching, the more accurately
is the long-time limit recovered. The present formalism can be
straightforwardly extended to infinite switch-on times. We show that this
yields highly accurate results for the long-time limit of both thermodynamic
observables and spectral functions, and overcomes the significant errors within
the single quench formalism [Anders {\em et al.}, Phys. Rev. Lett. {\bf 95},
196801 (2005); Nghiem {\em et al.}, Phys. Rev. Lett. {\bf 119}, 156601 (2017)].
This improvement provides a first step towards an accurate description of
nonequilibrium steady states of quantum impurity systems, e.g., within the
scattering states NRG approach [Anders, Phys. Rev. Lett. {\bf 101}, 066804
(2008)].
