One way to deal with the storage problem for the forward source wavefield in reverse time migration and full-waveform inversion is the reconstruction of that wavefield during reverse time stepping along with the receiver wavefield. Apart from the final states of the source wavefield, this requires a strip of boundary values for the whole time range in the presence of absorbing boundaries. The width of the stored boundary strip, positioned in between the interior domain of interest and the absorbing boundary region, usually equals about half that of the finite-difference stencil. The required storage in 3D with high frequencies can still lead to a decrease in computational efficiency, despite the substantial reduction in data volume compared with storing the source wavefields at all or at appropriately subsampled time steps. We have developed a method that requires a boundary strip with a width of just one point and has a negligible loss of accuracy. Stored boundary values over time enable the computation of the second and higher even spatial derivatives normal to the boundary, which together with extrapolation from the interior provides stability and accuracy. Numerical tests show that the use of only the boundary values provides at most fourth-order accuracy for the reconstruction error in the sourcewavefield. The use of higher even normal derivatives, reconstructed from the stored boundary values, allows for higher orders as numerical examples up to order 26 demonstrate. Subsampling in time is feasible with highorder interpolation and provides even more storage reduction but at a higher computational cost.