We perform the first systematic search of all NICER archival observations of black hole (and candidate) low-mass X-ray binaries for signatures of reverberation. Reverberation lags result from the light travel time difference between the direct coronal emission and the reflected disk component, and therefore their properties are a useful probe of the disk-corona geometry. We detect new signatures of reverberation lags in eight sources, increasing the total sample from three to 11, and study the evolution of reverberation lag properties as the sources evolve in outbursts. We find that in all of the nine sources with more than one reverberation lag detection, the reverberation lags become longer and dominate at lower Fourier frequencies during the hard-to-soft state transition. This result shows that the evolution in reverberation lags is a global property of the state transitions of black hole low-mass X-ray binaries, which is valuable in constraining models of such state transitions. The reverberation lag evolution suggests that the corona is the base of a jet that vertically expands and/or gets ejected during state transition. We also discover that in the hard state, the reverberation lags get shorter, just as the quasiperiodic oscillations (QPOs) move to higher frequencies, but then in the state transition, while the QPOs continue to higher frequencies, the lags get longer. We discuss the implications of the coronal geometry and physical models of QPOs in light of this new finding.