I've been with EE for years on a one month rolling contract, SIM only. I can get a much better with Smarty who use the EE network. I understand that I need to give EE 30 days notice and ask for the PAC so that I can keep the same number.
What I need help with is the new SIM from Smarty. Ideally I'd like to activate on the day that my EE service stops. Can I order the SIM now and activate it in 30 days time? I don't understand properly how these things work
Transferring your number to your new provider will not happen immediately. It usually takes one or two days. This is mentioned in the link that Corbyloon has provided. I would also ensure that you don't have any time remaining on your contract. I used the Three network for some time, but found it very patchy. Have been with Tesco for some time now and find the cost and service excellent.
If you particularly want to stay on the EE network, have a look at Plusnet. Both companies are owned by BT, but Plusnet's deals are cheaper than BT or EE.