Admin Mod reacts to user initiated events. As far as admin is concerned the server starting is not a user initiated event which means that Admin is not started when the server is started
Do not confuse loading with starting. Admin Mod is loaded when the server starts but it takes no actions apart from its load time procedures. It does not read its config files or load any of its plugins. When admin starts it reads the config files and loads the plugins. There are two separate events.
Admin Mod can be made to load when the server starts by placing an admin command in the load order. Make sure that the command will be issued well after the Admin mod specific configuration file is executed or you will recieve an error message.
The best command to use is considered to be admin_version. So the server.cfg would require the line:
admin_command admin_version
Most of the known steam mods no longer execute the server.cfg on mapchange. If you want to start
Admin Mod on mapchange you will need to force a config file to be executed on map changes. This is done by setting the cvar mapchangecfgfile to server.cfg and then follow the instructions above.
You can also create a new file like mapchange.cfg with admin_command admin_version inside and set mapchangecfgfile mapchange.cfg in the server.cfg. That way you may avoid the possibly overwriting cvars which were set in server.cfg during the mapchange while still automatically start Admin Mod.