Nginx subdomain routing during DNS propagation when switching data center - http

Question

I have three servers: One serving data through an API with IP address 11.111.11.11, one serving static resources with IP address 22.222.22.22 and one Nginx reverse proxy with IP address 33.333.33.33 routing data to the other two.
I have 3 type A DNS records for example.com, www.example.com and api.example.com all pointing to the Nginx server with IP address 33.333.33.33. It then redirects the requests either to itself or to one of the other two servers depending on the protocol and subdomain in the request. The Nginx server blocks look like this:
# Redirect static request to HTTPS with no subdomain
server {
listen 80 default_server;
server_name example.com www.example.com;
return 301 https://example.com$request_uri;
}
# Redirect API request to HTTPS
server {
listen 80;
server_name api.example.com;
return 301 https://api.example.com$request_uri;
}
# Redirect API request to same without subdomain
server {
listen 443 ssl;
server_name www.example.com;
return 301 https://example.com$request_uri;
}
# Route to static server
server {
listen 443 ssl;
server_name example.com;
### SSL conf...
location / {
proxy_pass http://22.222.22.22:8080;
}
}
# Route to API server
server {
listen 443 ssl;
server_name api.example.com;
### SSL conf...
location / {
proxy_pass http://11.111.11.11:8080;
}
}
Pretty basic stuff.
Now I have been for forced to switch data center. I now have a copy of the exact same setup at the new data center, but the IP addresses are now instead 44.444.44.44, 55.555.55.55 and 66.666.66.66.
Now comes the tricky part. I now want to update the DNS records (which can take up to 24h) to all point to the Nginx server with IP address 66.666.66.66 in the new data center without downtime.
To enable this I'm thinking of changing the configuration of the old Nginx server so that it routes all traffic to the new Nginx server. However, this will (probably) not work, since the routing is done using the request subdomains, and that information is lost in the redirect. I am unable to route traffic from the old Nginx server to the new API and static servers, since the servers talk to each other over a private network. The traffic also has to keep its encryption when on the internet, and can only be sent decrypted within the same data center.
How can I do this data center switch, while having no down time during the DNS propagation?