For the big disasters, as well as those en route to becomming big disasters, I tend to use the Imperial Ministry of Health and Sophont Services. They're patterned on the World Health Organization in terms of prevention and 'elevating health and saftey standards' (i.e. tracking down the source of virii, stopping plagues before they get too far along). But I gave them a darker side, for those times when things like 'Space Ebola' or bioweapons threaten. A few carefully modified vessels that move in and are capable of 'sanitizing' a site should things get that far. Attached to these ships are a few platoons of 'Medical Heavy Support Personnel' (Read also Marines and their ilk) to make sure that everyone who can get off the planet that either needs to or has to does.
The PCs have learned that when the HSS advisors and doctors pull out, and the big blue ships start moving in, its time to run. Leave the cargo, leave the money, leave the passengers behind no matter how much they're willing to pay.
So far, the HSS Operations Group Epsilon has only been called out and used once in the past century. But everyone in the frontier/scout medical community understands that they're out there, just in case.
As for dealing with disaster recovery, I tend to use the IISS to go in and act as advisors to the local governments. They tend to serve in the organizational and distribution networks. More like the Peace Corps and Red Cross than anything else.