As you expected, using nginx as a proxy is probably the best way to manage the Jenkins URL. Just setting it in the config won't work; Jenkins will still be listening on the configured port. Port defaults to 8080 but you can try chaning that to port 80. This post (while dated) gives some details on How to configure Jenkins to run on port 80.
I've always used the nginx option and its never failed me. You even get the benefit of being able to add SSL certificates. I recommend using let's encrypt.
Here's the configuration I've used for years, updated with your jenkins.HOST.com
URL. Just install Jenkins as usual, fix any selinux issues (either disable it or configure access for nginx to access localhost port 8080) and you should be good to go!
upstream jenkins {
server 127.0.0.1:8080;
}
server {
listen 80 default_server;
listen [::]:80;
server_name jenkins.HOST.com;
return 301 https://jenkins.HOST.com$request_uri;
}
server {
listen 443 ssl default_server;
listen [::]:443 ssl ipv6only=on;
server_name jenkins.HOST.com;
location / {
proxy_pass http://jenkins;
proxy_set_header Host $host;
proxy_set_header X-Forwarded-For $remote_addr;
}
ssl_certificate /etc/letsencrypt/live/jenkins.HOST.com/fullchain.pem;
ssl_certificate_key /etc/letsencrypt/live/jenkins.HOST.com/privkey.pem;
include /etc/letsencrypt/options-ssl-nginx.conf;
ssl_dhparam /etc/letsencrypt/ssl-dhparams.pem;
}