Transmitting messages in mobile wireless networks typically involves on-demand route discovery implemented via network-wide broadcast. Due to the dynamic nature of the network topology the life-time of a route is very short, so a source frequently requires a new route to an old destination. Simultaneous discovery of multiple routes can reduce the overhead due to repeated route discovery broadcasts. Previously proposed multipath protocols do not guarantee discovery of alternative paths if they exist. We propose a multiple route discovery algorithm (ALTDSR) that finds a (multihop) primary path between a source and a destination, and a set of alternative paths. We introduce dominator relationships between primary and non-primary path nodes. Using dominators, we characterize alternative paths that bypass an intermediate node on the primary path. We develop algorithms that guarantee finding a set of alternative paths to tolerate any single node fault on the primary path, if such a set of alternative paths exists. We present simulation results which show that under mobility conditions, ALTDSR delivers substantially more packets (around 75% more) than dynamic source routing (DSR) with moderate increase in routing overhead.