  A crucial component to maximizing the science gain from the multi-messenger
follow-up of gravitational-wave (GW) signals from compact binary mergers is the
prompt discovery of the electromagnetic counterpart. Ideally, the GW detection
and localization must be reported early enough to allow for telescopes to slew
to the location of the GW-event before the onset of the counterpart. However,
the time available for early warning is limited by the short duration spent by
the dominant ($\ell = m = 2$) mode within the detector's frequency band.
Nevertheless, we show that, including higher modes - which enter the detector's
sensitivity band well before the dominant mode - in GW searches, can enable us
to significantly improve the early warning ability for compact binaries with
asymmetric masses (such as neutron-star-black-hole binaries). We investigate
the reduction in the localization sky-area when the $\ell = m = 3$ and $\ell =
m = 4$ modes are included in addition to the dominant mode, considering typical
slew-times of electromagnetic telescopes ($30-60$ sec). We find that, in LIGO's
projected "O5" ("Voyager") network with five GW detectors, some of the
neutron-star-black-hole mergers, located at a distance of $40$ Mpc, can be
localized to a few hundred sq. deg. $\sim 45$ sec prior to the merger,
corresponding to a reduction-factor of $3-4$ ($5-6$) in sky-area. For a
third-generation network, we get gains of up to 1.5 minutes in early warning
times for a localization area of $100$ sq. deg., even when the source is placed
at $100$ Mpc.
