Different rule sets handle a thrown bat differently. But none that I am aware of call this an automatic out.
For high school ball (NFHS/FED) if a bat is carelessly discarded, the umpire is to issue a team warning. The next offender on that team that carelessly discards a bat is removed from the game and restricted to the dugout (a less severe penalty than ejection, since it does not carry a suspension from any subsequent games).
If a high school player intentionally throws a bat in anger, they are to be ejected, unless the umpire judges the infraction to be minor in nature.
But neither case calls for an "out". The offending player leaves the game and a substitute takes her place.
Someone posted the USSSA rule, which seems to be the same as the high school rule, and I will take their word for it since I do not call any USSSA ball.
The ASA Point-of-Emphasis has also been posted. This POE covers a carelessly thrown bat, which carries no penalty in ASA ball. A bat thrown intentionally, in anger, is grounds for ejection in ASA. Here again, neither of these infractions is an automatic out.
The NSA rule on thrown bats is the same as the ASA rule.
That a thrown bat is an automatic out is one of those rule myths that just won't go away. Some local leagues will add this rule, but you won't find it under the standard rules of softball.
One thing I have never seen in a league that calls this an "out" is an explanation in their rule that also specifies this to be a dead ball. The rule book lists all of the situations where the ball is dead and runners cannot advance. If your special league rules do not specifically call this a dead ball, then play is live and all other runners would be free to advance.
Just another example of what can happen when a league adds their own rules. Almost always, you inadvertantly affect another rule and create a potential conflict.