You need to automate everything you can with monitoring - manually rolling out agents, check's and alerting configuration is not supportable.
So really you want agents/instrumentation rolled into your build process, and that same process also making any required updates to your monitoring config to pickup new endpoints to scrape etc.
E.G if you use Kubernetes or say OpenStack, get a script to update a configuration file for prometheus in Git or some other SCM tool at the moment of deployment, and have something like Jenkins watching the repo and doing the deploy of configuration file to the prometheus box for you.
Similar stuff can be done for Nagios with its hosts.cfg and other associated files, or with Icinga you have your build process call the rest API to add the host into monitoring.
Obviously will all depend hugely on the size of your environment, the technologies used and probably most importantly the culture and expertise available in your organisation. It takes a bit of re-education and shift in mindset for people from a traditional ITIL/ITSM type support model moving into this new style of agile/devops/modern/new-style stuff (delete as appropriate!)