Continuous gravitational waves from spinning deformed neutron stars have not been detected yet, and are one of the most promising signals for future detection. All-sky searches for continuous gravitational waves from unknown neutron stars in binary systems are the most computationally challenging search type. Consequently, very few search algorithms and implementations exist for these sources, and only a handful of such searches have been performed so far. In this paper, we present a new all-sky binary search method, BinarySkyHouF, which extends and improves upon the earlier BinarySkyHough method, and which was the basis for a recent search [Covas et al. [1]]. We compare the sensitivity and computational cost to previous methods, showing that it is both more sensitive and computationally efficient, which allows for broader and more sensitive searches.