As Jelv says you do not need a new phone to go pay monthly, its usually an expensive way to buy a new phone, and more often than not you're locked in to a 24 month contract. Often the price will not drop after the contract ends to account for the fact the phone is paid off, but I think new regulations are coming in. If changing networks and keeping your existing phone just be careful its not locked to your current network.
All providers do pay monthly SIM only deals, I've always purchased my phones outright and used pay monthly.
It's very easy to transfer your phone number between providers, you can get a PAC (Port Authorisation Code) code via text now,just text PAC to 65075 regardless of network -
see hereYou give the PAC code to your new provider, and they transfer your number.
Moving a number from PAYG to pay monthly whilst staying with the same carrier can sometimes be a bit trickier but Three seems to have it covered -
see hereI have heard that IdMobile and Smarty don't get the full speed of the Three network, not sure how true this is though.
If staying with Three they do some great deals (although ID & Smarty will be cheaper), but you need to look around for them, then go into a Three store with proof and ask them to match it. I'm on Three with 8GB data, unlimited minutes and texts for £10 a month, although the cheapest I've seen that for is £8, but the store couldn't go that low - I got a free months, so effectively around £9 a month.