In wireless sensor networks, real-time data delivery schemes typically achieve the desired delivery speed by proactively performing one-hop lookahead. Recently, to reduce the deadline miss ratio with respect to the desired delivery speed, a study has proposed a real-time data delivery scheme based on proactively performing two-hop lookahead. However, the recent proposal might cause heavy message exchange overhead and high computing complexity in order to proactively obtain two-hop neighbor speed information in all sensor nodes whether data are delivered or not. In this paper, we propose a novel real-time data delivery scheme that applies on-demand multi-hop lookahead only around data forwarding paths. Hence, the scheme can provide lower deadline miss ratio for real-time data delivery with low message exchange overhead than existing schemes.