  While binary pulsar systems are fantastic laboratories for a wide array of
astrophysics, they are particularly difficult to detect. The orbital motion of
the pulsar changes its apparent spin frequency over the course of an
observation, essentially "smearing" the response of the time series in the
Fourier domain. We review the Fourier domain acceleration search (FDAS), which
uses a matched filtering algorithm to correct for this smearing by assuming
constant acceleration for a small enough portion of the orbit. We discuss the
theory and implementation of a Fourier domain "jerk" search, developed as part
of the \textsc{PRESTO} software package, which extends the FDAS to account for
a linearly changing acceleration, or constant orbital jerk, of the pulsar. We
test the performance of our algorithm on archival Green Bank Telescope
observations of the globular cluster Terzan~5, and show that while the jerk
search has a significantly longer runtime, it improves search sensitivity to
binaries when the observation duration is $5$ to $15\%$ of the orbital period.
Finally, we present the jerk-search-enabled detection of Ter5am
(PSR~J1748$-$2446am), a new highly-accelerated pulsar in a compact, eccentric,
and relativistic orbit, with a likely pulsar mass of
1.649$^{+0.037}_{-0.11}$\,\msun.
