Something I found from mlb.com in a section called Ask The Umpire:
A batter beats a close play at first and turns toward foul territory after crossing the bag and running it out. However, the throw gets misplayed and upon seeing the ball loose, the batter breaks for second. Does he have to come back and retouch first base, before continuing on to second? It seems that you can over run first base if you turn toward foul territory. If you turn toward second, you are a base runner, available to advance and be put out. This play happens a lot in unprofessional leagues, and the rule is not clear. Thank you.
The answer to the first part of your question is no, the batter does not have to go back and retouch first base (provided he touched it to begin with!). He may continue on toward second base at any point after overrunning first.
Regarding the second part of your question, it really does not matter which way the player turns after overrunning first base! The key under the rule -- Rule 7.08(c) -- is that the batter-runner "returns immediately to the base." Any attempt of the batter-runner to advance to second is considered not returning immediately, thereby putting the runner in jeopardy of being tagged out. This usually happens when the runner, thinking the ball has been overthrown far enough that he can make it to second, takes a quick step or jerk toward second, thereby forfeiting his right to return to first base safely. Such a runner can be tagged out.
I just care about the part that states 'it really does not matter which way the player turns after overrunning first base!' I haven't checked the official softball rules, but I'm sure it's the same.
So in our must-win church league softball game this weekend, I hit a grounder to the shortstop and used my speed, which gets a 94 in the EA Sports ratings, to beat the throw to first. I hear the ump call safe as I ran past the bag, so I turn around to take my position on first base. Then the first baseman says something to the umpire, tags me with the ball and now I'm called out. I guess I wasn't paying attention during the practice where Jeff taught people to run through the bag at first and then turn right into foul territory.
According to my knowledge of baseball rules (in this case, Rule 7.08), as long as I did not make a step or motion toward second base or the dugout, I could safely return to first. I was running along the baseline in foul territory, turned around and walked straight back toward first. It appears that I pivoted left as I headed back to my hard earned base and that somehow made me a live runner again to be tagged out. I protested because I did not make a move toward second. One of the other umpires overruled the first base ump and I stayed on base.
But the rest of the game, we had batter-runners on our team running down the line after a grounder and veering way right after the bag. Everyone kept saying 'turn right' and we almost used that as our team cheer thing. I thought turning left was okay and I'm glad that major league umpires agree.
In my defense, I usually hit the ball much further so I'm used to just jogging around the bases rather than having to run hard to first. The Greg Anderson training program really works. Just kidding. I only have warning track power. (EA Sports NCAA MVP Baseball 2006 power rating of just 71.)
Our team did win that game as we swept a doubleheader today, which should put us in the playoffs.