He is, 'to put it mildly', a demanding boss. [=he is a very demanding boss]
He's 'mildly' [='slightly, somewhat'] interested in politics.
She was upset, 'to put it mildly'. [=she was extremely upset]
(..) after the rag was taken out? It would have to be a very serious condition. Is it possible to have a serious coronary condition... where the warning signals were so mild as to escape a physician... during a routine medical exam? Possibly. There would still be symptoms, though. (..)1992 A Few Good Men