A few days before your cutover to the new IP address:
- Lower the TTL on your DNS records, so that when you DO make the change, it is updated around the DNS system quickly. A TTL of 10mins or so is a good tradeoff
- adding the new IP to your existing site so that it can run on both the old and new. You mention SSL, so I'll assume it's a web server of some sort, which can normally be configured pretty simply to serve on multiple IPs
A few days later (or basically when you're sure the original TTL on the DNS record has expired and DNS servers now have your lower TTL cached), update the DNS record to become your new IP. You should start to see traffic via your new IP pretty quickly.
Leave it this way for a few days as DNS caches expire and pickup your new IP and start using it. Once you're happy that you're seeing a normal level of traffic via your new IP, remove the old one from your server.
There really isn't a quicker way to do it without experiencing some level of interruption to users.