From the first phase of the high-cadence Formation of the Outer Solar System: an Icy Legacy (FOSSIL) survey, we analyzed lightcurves, ranging from one to four nights in length, of 371 trans-Neptunian objects (TNOs) for periodicity. We found 29 TNOs with periodic lightcurves, one of which is a good candidate for a close/contact binary. Another of the periodic FOSSIL TNOs could potentially have the fastest of all known TNO spin rates, with a period of 1.3hr. We do not have total confidence in the period and thus plan to obtain a more detailed lightcurve for confirmation. The periodic TNOs have an average rotation period of 11.2hr, close to the value obtained by Alexandersen+ (2019, J/ApJS/244/19), which had similar cadence, but different from other surveys. In regards to contention in the literature about whether smaller TNOs are more irregular in shape and thus have larger lightcurve amplitudes, we found that there is a weak correlation between absolute magnitude and lightcurve amplitude in a subset of 194 FOSSIL TNOs, even when using the more appropriate brightest (minimum) absolute magnitude instead of the time-averaged value.