As a matter of fact he should take leading role during any problems raised during emergencies. When a Chief Minister resigns and if the Governor asks him to continue till the next arrangement is made to continue he will continue as care taker. In case of elections, till the results comes the CM will be a regular CM only but not a caretaker.
Thoughts exchanged is knowledge gained.
"If you are killed in action, you go to Heaven. If you win, you rule this Earth (as beautiful as Heaven). That is why, O son of Kunti, take a firm resolve and fight!"-- Shrimad Bhagwad Gita
Even at the level of Prime Minister, sometimes such an arrangement is done and in that case the President appoints an interim Prime Minister till a regular one is selected. For example after the death of Jawaharlal Nehru, the President of India appointed Gulzarilal Nanda as the Interim Prime Minister. He remained in that capacity till the regular Prime Minister Lal Bahadur Shastri was sworn in.
Knowledge is power.