But, in HS, s/he must leave live-ball territory to be out. ...
Not true.
NFHS Case Play 8.6.22 (2010) 8.6.22 SITUATION A: B1 reaches first safely. In her overrun, she breaks for second and then "gives up" far away from the baseline. RULING: If B1 is attempting to reach a base, she must be tagged out, but the fielder is not expected to chase her into the outfield. In the outlined case, B1 is out for giving up. (8-4-2; 8-6-2)