Before you even START looking, read and understand the early-termination provisions in your lease, if there are any. Is there a an early-termination fee? When there is one, a fee of two months rent to buy your way out is common.
You need to know this. Because, if you start looking, you just may find the place you just can't live without, and if you don't take it then, it won't be available when you want it. If there is a buyout clause in your lease and you can afford that buyout fee, you can move when you want.
If that is not an option, I suggest starting your search about April 1st, maybe a week sooner. This gives you 4-5 weeks to find a place that's suitable and still be able to give your current landlord the required notice that you will be vacating when the lease expires. You must do this before the end of April, give notice. Hopefully you can have the new place beginning June 1st. If you find something in April and you have to take it May 1st, then you will be paying double rent for one month.
The only way to avoid paying double rent is to make that leap of faith and go ahead and give notice that you will be leaving by May 31st and start looking around the end of April and hope you find something suitable to occupy June 1st.