We report the first measurement of target single-spin asymmetries (A$_N$) in the inclusive hadron production reaction, $e~$+$~^3text{He}^{uparrow}rightarrow h+X$, using a transversely polarized $^3$He target. The experiment was conducted at Jefferson Lab in Hall A using a 5.9-GeV electron beam. Three types of hadrons ($pi^{pm}$, $text{K}^{pm}$ and proton) were detected in the transverse hadron momentum range 0.54 $<p_T<$ 0.74 GeV/c. The range of $x_F$ for pions was -0.29 $<x_F<$ -0.23 and for kaons -0.25 $<x_F<$-0.18. The observed asymmetry strongly depends on the type of hadron. A positive asymmetry is observed for $pi^+$ and $text{K}^+$. A negative asymmetry is observed for $pi^{-}$. The magnitudes of the asymmetries follow $|A^{pi^-}| < |A^{pi^+}| < |A^{K^+}|$. The K$^{-}$ and proton asymmetries are consistent with zero within the experimental uncertainties. The $pi^{+}$ and $pi^{-}$ asymmetries measured for the $^3$He target and extracted for neutrons are opposite in sign with a small increase observed as a function of $p_T$.
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