For the obsessive compulsive like me you can check on the Hassio github page https://github.com/home-assistant/hassio when the new Home Assistant version is available (look up for changes in the version.json file).
As soon as it changes, you can reboot Hassio to force the version check and the Update button will appear.
In an attempt to be proactive and do something about this, I have created a PR for the page that talks about updating Home Assistant designed to let Hass.io users know what’s happening.