Getting caught doesn't mean your run is over anymore!
Instead of respawning immediately, you'll now be sent to a Mini Jail. Your digging progress will still be safe.
To escape the mini jail and continue the digging, use a Keycard, which can be obtained from Prisoners. Every player starts each game with 1 free Keycard, giving you one opportunity to break out and continue digging from where you were caught.
Use it wisely—this extra chance could be the difference between escaping and starting over!