I updated to 2023.6.1 on my supervised installation. Now I woke up twice already to my HA not running anymore. Restarting the Raspi fixed the issues both time. If you run HA supervised, I advice you to wait with the update for some time. For me, I will take this as a lesson and go back to HAOS
I had to look it up because I wasn't sure. According to what I found you should enable logger in your configuration.yaml
The default is to log warnings, you'll want to put:
logger
default: info
Or something similar to the above. Then when it crashes you'll want to take the SD card to another computer before HA boots back up and look at the /config/home-assistant.log file. On boot it clears that file is why you want to read it before it boots again.
At 4AM this morning the dog woke me up. I noticed that my night light automations weren't working and so quickly checked HA -- found that my recently rebuilt server (Debian) had crashed, not just HA. I couldn't ping or SSH and so had to force a power down/reboot.
Not sure if it's the same symptom as yours, but that's never happened to me before.