My employer provides me with health insurance coverage. Is it advisable to take another policy on my own ?

If the sum assured is good and the company is doing fine then you should think positive and do not go in for additional sum. When you plan to leave the organization then you must verify from the new employer whether they have suitable sum assured available for you as part of compensation package. If no health insurance is available then you must buy Health Insurance immediately so that 30 days cooling off period of new policy (purchased by you) is matching with last 30 days with your employer. Try to have this health insurance from the same Insurance Company.