  Modelling the end point of binary black hole mergers is a cornerstone of
modern gravitational-wave astronomy. Extracting multiple quasinormal mode
frequencies from the ringdown signal allows the remnant black hole to be
studied in unprecedented detail. Previous studies on numerical relativity
simulations of aligned-spin binaries have found that it is possible to start
the ringdown analysis much earlier than previously thought if overtones (and
possibly mirror modes) are included. This increases the signal-to-noise ratio
in the ringdown making identification of subdominant modes easier. In this
paper we study, for the first time, black hole binaries with misaligned spins
and find a much greater variation in the performance of ringdown fits than in
the aligned-spin case. The inclusion of mirror modes and higher harmonics,
along with overtones, improves the reliability of ringdown fits with an early
start time; however, there remain cases with poor performing fits. While using
overtones in conjunction with an early ringdown start time is an enticing
possibility, it is necessary to proceed with caution. We also consider for the
first time the use of numerical relativity surrogate models in this type of
quasinormal mode study and address important questions of accuracy in the
underlying numerical waveforms used for the fit.
