you still keeps money in savings account? I only keep a little money in savings account, rest is invested. You can do this or if you don't feel safe in investing then yeah you can open another savings account. You can open as many as you want, doesn't matter all are safe.
I think you should try Auto-Sweep Facility in one savings account which you don't use regularly. Then the second one, you can use easily for daily transaction. It might help you to handle your money well