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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.