IL passed a law that recognizes such cases and allows automatic recognition in cases where aid is requested, due to an event like a riot or disaster overwhelming local resources.
And IIRC, after Katrina similar legislation was discussed on a federal level, to allow aid from other states to assist in disasters. Not sure if it passed or not.
Of course, back in the day it was commonplace for the SD to deputize local LEO's, to avoid interjurisdictional

, especially in cases of "mutual aid". (Issued 'em neat ID cards & everything

) A side effect was the way the ILCS was written, a deputy in IL was an "auxiliary" ("adjunct", or similar language) State Patrol officer, and thus had jurisdiction anywhere in the state.
Not sure if that's still true, but..