  We present a new rest-frame color-color selection method using "synthetic
$u_s-g_s$ and $g_s-i_s$'', $(ugi)_s$ colors to identify star-forming and
quiescent galaxies. Our method is similar to the widely-used $U-V$ versus $V-J$
($UVJ$) diagram. However, $UVJ$ suffers known systematics. Spectroscopic
campaigns have shown that $UVJ$-selected quiescent samples at $z \gtrsim 3$
include $\sim 10-30\%$ contamination from galaxies with dust-obscured star
formation and strong emission lines. Moreover, at $z>3$, $UVJ$ colors are
extrapolated because the rest-frame J-band shifts beyond the coverage of the
deepest bandpasses at $< 5~\mu m$ (typically $Spitzer$/IRAC 4.5 $\mu m$ or
future $JWST$/NIRCam observations). We demonstrate that $(ugi)_s$ offers
improvements to $UVJ$ at $z>3$, and can be applied to galaxies in the $JWST$
era. We apply $(ugi)_s$ selection to galaxies at $0.5<z<6$ from the (observed)
3D-HST and UltraVISTA catalogs, and to the (simulated) JAGUAR catalogs. We show
that extrapolation can affect $(V-J)_0$ color by up to 1 magnitude, but changes
$(u_s-g_s)_0$ color by $\leq$ 0.2 mag, even at $z\simeq 6$. While
$(ugi)_s$-selected quiescent samples are comparable to $UVJ$ in completeness
(both achieve $\sim$85-90% at $z=3-3.5$), $(ugi)_s$ reduces contamination in
quiescent samples by nearly a factor of two, from $\simeq$35% to $\simeq$17% at
$z=3$, and from $\simeq $60% to $\simeq $33% at $z=6$. This leads to
improvements in the true-to-false-positive ratio (TP/FP), where we find TP/FP
$\gtrsim$ 2.2 for $(ugi)_s$ at $z \simeq 3.5 - 6$, compared to TP/FP $<$ 1 for
$UVJ$-selected samples. This indicates that contaminants will outnumber true
quiescent galaxies in $UVJ$ at these redshifts, while $(ugi)_s$ will provide
higher-fidelity samples.
