People imagine a website migration as a switch being thrown, with a nervous gap in the middle where the site is nowhere. It does not have to work like that, and when it does, it is because the order was wrong.
Build First, Point Second
The new host can have a complete, working copy of your site before your domain knows anything about it. Files copied, database imported, SSL issued, the lot.
You test that copy using a temporary address or by overriding DNS on your own machine. You click through the pages, submit a form, log into the admin, check the shop. If something is broken you fix it while the live site carries on serving visitors from the old host, untouched.
Only when the copy is genuinely right do you change where the domain points.
Lower The TTL Beforehand
Every DNS record has a time to live. How long other servers may cache it. If yours is set to a day, some visitors will keep reaching the old host for a day after you switch.
Drop it to five minutes a day before you move. Then the changeover propagates in minutes rather than overnight, and you can put it back afterwards.
Leave The Old One Running
Do not cancel the old hosting the moment DNS changes. Leave it up for a week. During the crossover both hosts serve the same site, which is exactly what you want, and if something turns out to be wrong you can point back instantly.
The one thing to watch during that window is anything writing data. If the shop takes an order while a visitor is still being served by the old host, that order is in the old database. For sites taking orders, do the final database copy right at the switch, or pause ordering for the few minutes it takes.
Email Is A Separate Job
The most common migration disaster is not the website at all. It is discovering that moving the domain's DNS also moved the mail records, and nobody planned for it.
Check where your email lives before touching DNS. If it is staying put, carry the MX records across exactly.
We do migrations for customers as part of moving in, and this is the sequence we follow. It is not clever. It is just done in the right order.