  Accurate waveform models describing the complete evolution of compact
binaries are crucial for the maximum likelihood detection framework, testing
the predictions of General Relativity (GR) and investigating the possibility of
an alternative theory of gravity. Deviations from GR could manifest in subtle
variations of the numerical value of the GW signal's post-Newtonian (PN)
phasing coefficients. Once the search pipelines confirm an unambiguous signal
detection, deviations of the signal phasing coefficients at various PN orders
are routinely measured and reported. As the search templates themselves do not
incorporate any deviations from GR, they may miss astrophysical signals
carrying a significant departure from general relativity. We present a
parametrized template-based search for exotic gravitational-wave signals beyond
General Relativity by incorporating deviations to the signal's phasing
coefficients at different post-Newtonian orders in the search templates. We
present critical aspects of the new search, such as improvements in search
volume and its effect on various parts of the parameter space. In particular,
we demonstrate a factor x2 increase in search sensitivity (at a fixed
false-alarm rate) to beyond-GR exotic signals by using search templates that
admit a range of departures from general relativity. We also present the
results from a re-analysis of the 10-days long duration of LIGO's O1 data,
including the epoch of the GW150914 event, highlighting the differences from a
standard search. We indicate several directions for future research, including
ways of making the proposed new search computationally more efficient.
