A batter-runner may OVERRUN first base and return directly to it without liability to be put out. 

If she attempts to ADVANCE toward second base and decides to RETURN (to first) she is at risk to be tagged out.

However once the PITCHER has control of the live ball WITHIN THE PITCHING CIRCLE the following items may also apply:

•  The BATTER-RUNNER may ROUND first base (towards 2nd base), STOP momentarily, then immediately WITHOUT STOPPING again …

•  RETURN to first base.

• OR she may ADVANCE to second base.

 Provided the BATTER-RUNNER stops ONCE momentarily then either ADVANCES or RETURNS, it shall be legal … but is at liability to be tagged out if the pitcher makes a play on her.

This includes a BATTER-RUNNER advancing from a walk, dropped 3rd strike, or any legal method while the ball is live.

For more information on rule differences between high school, amateur, and independent softball organizations ... CLICK HERE.

Rules Reference
NFHS 8-7-2; 8-6-8; 8-7-4 / ASA 8-7T(3); RS #37 / USSSA 8.2; 8.10
NSA 8-8i / PONY 9-8t  / BABE 7.13; NR