If you want to be able to handle an error instead of blindly exiting, instead of using set -e, use a trap on the ERR pseudo signal.
#!/bin/bash
f () {
    errorCode=$? # save the exit code as the first thing done in the trap function
    echo "error $errorCode"
    echo "the command executing at the time of the error was"
    echo "$BASH_COMMAND"
    echo "on line ${BASH_LINENO[0]}"
    # do some error handling, cleanup, logging, notification
    # $BASH_COMMAND contains the command that was being executed at the time of the trap
    # ${BASH_LINENO[0]} contains the line number in the script of that command
    # exit the script or return to try again, etc.
    exit $errorCode  # or use some other value or do return instead
}
trap f ERR
# do some stuff
false # returns 1 so it triggers the trap
# maybe do some other stuff
Other traps can be set to handle other signals, including the usual Unix signals plus the other Bash pseudo signals RETURN and DEBUG.