The conflict in Ukraine is causing large-scale displacement in Europe and in the World. Based on the United Nations High Commissioner for Refugees (UNHCR) estimates, more than 7 million people fled the country as of 5 September 2022. In this context, it is extremely important to anticipate where these people are moving so that national to local authorities can better manage challenges related to their reception and integration. This work shows how innovative data from social media can provide useful insights on conflict-induced migration flows. In particular, we explore the potential of Facebook's Social Connectedness Index (SCI) for predicting migration flows in the context of the war in Ukraine, building on previous research findings that the presence of a diaspora network is one of the major migration drivers. To do so, we first evaluate the relationship between the Ukrainian diaspora and the number of refugees from Ukraine registered for Temporary Protection or similar national schemes as a proxy of migratory flows into the EU. We find a very strong correlation between the two (Pearson's r=0.94, p<0.0001), which indicates that the diaspora is attracting the people fleeing the war, who tend to reach their compatriots, in particular in the countries where the Ukrainian immigration was more a recent phenomenon. Second, we compare Facebook's SCI with available official data on diaspora at regional level in Europe. Our results suggest that the index, along with other readily available covariates, is a strong predictor of the Ukrainian diaspora at regional scale. Finally, we discuss the potential of Facebook's SCI to provide timely and spatially detailed information on human diaspora for those countries where this information might be missing or outdated, and to complement official statistics for fast policy response during conflicts.