  We present multi-band Period-Luminosity (PL) relations for first-overtone
mode Cepheids in the Small Magellanic Cloud (SMC). We derive optical band PL
relations and the Wesenheit function using $VI$ mean magnitudes from the
Optical Gravitational Lensing Experiment (OGLE-III) survey. We cross-match
OGLE-III first-overtone mode Cepheids to the 2MASS and SAGE-SMC catalogs to
derive PL relations at near-infrared ($JHK_s$) and mid-infrared
($3.6~\&~4.5\mu\mathrm{m}$) wavelengths. We test for possible non-linearities
in these PL relations using robust statistical tests and find a significant
break only in the optical-band PL relations at 2.5 days for first-overtone mode
Cepheids. We do not find statistical evidence for a non-linearity in these PL
relations at 1 day. The multi-band PL relations for fundamental-mode Cepheids
in the SMC also exhibit a break at 2.5 days. We suggest that the period break
around 2.5 days is related to sharp changes in the light curve parameters for
SMC Cepheids. We also derive new optical and mid-infrared band PL relations for
first-overtone mode Cepheids in the Large Magellanic Cloud (LMC). We compare
multi-band PL relations for first-overtone mode Cepheids in the Magellanic
Clouds and find a significant difference in the slope of the $V$-band PL
relations but not for $I$-band PL relations. The slope of PL relations are
found to be consistent in most of the infrared bands. A relative distance
modulus of $\Delta\mu=0.49\pm0.02$~mag between the two clouds is estimated
using multi-band PL relations for the first-overtone mode Cepheids in the SMC
and LMC.
