Irish people left voluntarily during the Great Famine (1845-1852) in large volume. Granted, their only two options were pretty much leave now or starve. But still, a choice existed.

Welsh migration to Patagonia (Y Wladfa), slightly later than you're after though (1865) and not really a huge number went, could be of interest though.

Although I'm not entirely sure about volume, Norse settlers established themselves in several different locations. Namely; Kiev, Normandy, the British Isles, Iceland and potentially other places I have forgotten to mention. They often installed themselves as lords and later assimilated into the local culture (Kiev and Normandy) but Iceland was uninhabited prior to Norse settlement and has preserved the Norse language to an impressive degree.

If we're going with "leave now or starve" as the criteria, than perhaps you may want to look into the Nahua migrations of Mexico. If Beekman and Christensen are right about the point of origin of Nahuatl speaking groups than these people either had to leave the Bajio or starve due to drought. The migrations of these people caused a ripple effect throughout Mesoamerica with some groups migrating as far as El Salvador.