Most often, skipping inward has to do with the point a which the arm is set down and the speed at which the arm is set down. Records have a raised bead around the edge of the record to strengthen the record at its most exposed part, and in the old days, to keep the playing surface from contacting another record when stacked for automatic play. When the needle is set down on top of that raised bead, it falls down that little hill and can gain enough momentum to slide past the first grooves. If you have the ability to adjust the point of set down, choose a point farther into the record and away from the edge. It seems counter intuitive to set a needle down further into the record when it is skipping the first grooves, but this works. It also helps to have a slow descent of the arm to also reduce the momentum sliding down the edge bead.
If you cannot adjust the set down point, you can keep the arm cued up, and before cuing down, you can nudge the arm a little bit inward to miss the bead.
While excessive anti-skating can exacerbate the problem, it is not the primary cause. If it were the cause the needle would continue to slide in well past the first few grooves because skating force is not much different between the first contact point and where it stops sliding inward.