  M-dwarfs are known to commonly host high-multiplicity planetary systems.
Therefore M-dwarf planetary systems with a known transiting planet are expected
to contain additional small planets ($r_p \le 4$ R$_{\oplus}$, $m_p \lesssim
20$ M$_{\oplus}$) that are not seen in transit. In this study we investigate
the effort required to detect such planets using precision velocimetry around
the sizable subset of M-dwarfs which are slowly rotating ($P_{\mathrm{rot}}
\gtrsim 40$ days) and hence more likely to be inactive. We focus on the test
case of GJ 1132. Specifically, we perform a suite of Monte-Carlo simulations of
the star's radial velocity signal featuring astrophysical contributions from
stellar jitter due to rotationally modulated active regions and keplarian
signals from the known transiting planet and hypothetical additional planets
not seen in transit. We then compute the detection completeness of
non-transiting planets around GJ 1132 and consequently estimate the number of
RV measurements required to detect those planets. We show that with 1 m
s$^{-1}$ RV precision per measurement, only $\sim 50$ measurements are required
to achieve a 50\% detection completeness to all non-transiting planets in the
system and to planets which are potentially habitable. Throughout we advocate
the use of Gaussian process regression as an effective tool for mitigating the
effects of stellar jitter including stars with high activity. Given that GJ
1132 is representative of a large population of slowly rotating M-dwarfs, we
conclude with a discussion of how our results may be extended to other systems
with known transiting planets such as those which will be discovered with TESS.
