Hosting is the environment your website lives on, plus the unglamorous work of keeping it there: SSL, backups, updates and someone to call when the site is down.
Shared budget hosting is fine for a personal page. It is a poor home for a site that generates enquiries. You are sharing resources, support is generic, and backups are easy to assume until you need one.
Before you move host, ask where backups are stored, how restores are tested, who manages updates, and what happens to email on the same domain. DNS mistakes take sites and inboxes down together.
If the same team designed the site and hosts it, they already know the stack. That is worth more than a marginally cheaper monthly plan from a company that has never opened the codebase.
If you are briefing a project in the South West, you may also want web design, website development or a conversation about the brief.