if your agreement was for 12 months and you want to move out at the end of it, which I think is what you're saying, you don't need to give notice, you can simply move out on 3 June as your agreement has ended.
If you want to stay on and you have no new agreement then you will be on a periodic tenancy. This has the same terms as your existing tenancy except you need only give 1 months notice to quit and the landlord has to give you 2 months notice to vacate.
If you want to get out asap then you can leave on 3 June without paying any more rent and without giving notice.