There are 2 methods, and they depend on whether or not you are using hosted website builders, where you have to use that company’s web hosting.
If you are switching from a hosted website builder like Wix, Weebly, or Squarespace, for example, build the new website on another server. You will also have to do this if you are switching to a hosted website builder. You will have to pay for 2 servers while your new website is getting built. However, If you are planning on using the same domain name for your new website and your new website builder is an online website builder, like most are, you will need to either quickly switch over your current website content over to the new website builder or use a temporary domain name for the new website and then switch over your domain name to it when ready.
If you are using a website builder that allows you to choose your web hosting and the new website builder does too – like WordPress, Magento, Joomla, UltimateWB – you can build your new website on another folder on the same server. Then you only have to pay for the one server, even if you want to switch servers for your new website. For example, when people want to switch from WordPress to UltimateWB, and also use the UltimateWB web hosting plan, we just have to upload the WordPress website to the new server, and then install UltimateWB on a subfolder. When the new website is ready, we just move the new website over to the root of the server.