  Previously we proposed a new approach of exciting the $^{229}$Th nucleus
using laser-driven electron recollision [W. Wang et al., Phys. Rev. Lett. 127,
052501 (2021)]. The current article is aimed to elaborate the method by
explaining further theoretical details and presenting extended new results. The
method has also been improved by adopting the electronic excitation cross
sections calculated recently by Tkalya [E. V. Tkalya, Phys. Rev. Lett. 124,
242501 (2020)]. The new cross sections are obtained from Dirac distorted-wave
calculations instead of from Dirac plane-wave calculations as we used
previously. The distorted-wave cross sections are shown to be 5 to 6 orders of
magnitude higher than the plane-wave results. With the excitation cross
sections updated, the probability of isomeric excitation of $^{229}$Th from
electron recollision is calculated to be on the order of $10^{-12}$ per nucleus
per (femtosecond) laser pulse. Dependency of the excitation probability on
various laser parameters is calculated and discussed, including the laser
intensity, the laser wavelength, and the laser pulse duration.
