Hot on Schofill's heels is LSU senior Ingrid Lindblad, who is just one back of the leader after posting a 3-under par 68 in the opening round.
LSU freshman Edit Hertzman found herself at 3-over over her first five holes but fought back with five birdies en route to 2-under par 69, which gives her a share of third place with Mathilde Claisse of South Carolina and Rosie Belsham of Baylor.
Junior Carla Tejedo's 1-over 72 and senior Latanna Stone's 3-over 74 rounded out LSU's scoring.
In addition to Claisse's 2-under 69, the host Gamecocks also benefitted from a 1-under 70 from freshman Mia Sandtorv Lussand while All-Americans Louise Rydqvist (71) and Hannah Darling (73) accounted for the final counting scores.
Schofill, ranked 17th in the world by Golfweek/AmateurGolf.com, made five birdies over her first nine holes to go out in 30, picked up another birdie on her 15th hole to move to 6-under and withstood bogeys on two of her final three holes for a round of 67.
"In my front nine, I made a lot of putts and hit the ball really well to get off to a hot start," Schofill said. "I struggled a little bit the last few holes and just managed to make pars. I felt like I hit it in the right places today on this golf course."
Starting her round on Long Cove's back nine, Belsham turned in a bogey-free 69, picking up birdies on holes Nos. 14 and 15. The Baylor junior finished with 12 straight pars to gain a share of third place.
Only eight golfers managed to break par on the first day on the 6,220-yard Long Cove layout.
Second-ranked Wake Forest is eight back and tied for third after the Demon Deacons carded a 7-over 291. No. 3 Oregon finds itself tied for seventh with Texas at 11-over 295.
The Darius Rucker Intercollegiate continues on Tuesday and Wednesday from Long Cove Club in Hilton Head, S.C. GOLF Channel will provide live coverage each day beginning at 2:30 p.m. ET.