The best solution for you, is to move to a new host with letsencrypt enabled on their web hosting. However you should make note that a small interruption of around 1-4 hours is still possible. Because lets encrypt ssl are alloted or generated based on IP, so when you change the dns to new server and point the A records to their hosting, then it takes around 30 minutes to 24 hours to get the dns propagation to complete. And you can generate letsencrypt certificate on new server once the dns is resolving at the server end. so a small interruption is possible.
However I am not sure, if exporting ssl from old host to new host will work or not in case of letsencrypt ssl certificates. Even it it gets exported and imported correctly, you may get issues during renewal. So I strongly recommend you to generate new LE SSL on your new host fresh. It makes sure that everything works perfectly.
Another solution:
If you want, you can use cloudflare Origin pull SSL certificate, which is offered for 10 years term, so you can install it on existing host and new host at the same time and this will not affect ssl interupption during migration. So this is last solution I have for you, if you are interested in cloudflare ssl. You can get letsencrypt ssl running, once you are moved to new host and everything settles after a few weeks.
Hope this helps you.