In time-frequency analysis, a common problem is the retrieval of precise instantaneous frequency (IF) information from signals with time-varying oscillatory components. Reassignment methods are often used for estimating IFs, as these methods circumvent the well-known Fourier uncertainty principle that limits the concentration quality of linear transforms such as the short-time Fourier transform and continuous wavelet transform. The synchrosqueezing transform (SST), where reassignment is performed exclusively in the frequency direction, has become widely used due to its mathematically rigorous formulation and its formula for direct reconstruction of signal components from SST coefficients. However, the SST as originally formulated is ineffective when the IFs vary quickly. In this dissertation, we construct a version of SST that accounts for fast IF variations using a quilted short-time Fourier transform (QSTFT), a signal-adaptive patchwork representation arising from STFTs computed with different window functions. We first build an analysis-resynthesis framework for QSTFT in depth, to show how a perfect reconstruction can be made from a multi-window signal analysis. We then establish new theoretical results showing that the usage of chirped windows enables the QSTFT and its respective SST (SST-QSTFT) to concentrate signal information effectively, and we give the proofs of both these new results and our previously published theoretical results on the SST-QSTFT. Finally, we apply the SST-QSTFT to several numerical experiments on a dataset of animal call recordings as well as synthetic data, demonstrating its effectiveness for isolating precise IF information.