how to start spline movement at point of contact?

[hybrid solstice]

I am trying to develop a rail grind mechanic using splines and having the player move along the spline. How can I make my player move along the spline at the point of contact?

        {
            t += Time.deltaTime * speed;

            railGrindSparks.gameObject.SetActive(true);

            playerMovement.getRigidBody2D().gravityScale = 0f;

            var splinePosition = splinePath.EvaluatePosition(t);

            var newPosition = new Vector3(splinePosition.x, splinePosition.y + offsetY, splinePosition.z);

            transform.position = newPosition;
        }
        else {
            DetatchFromRail();
        }```

[clear anvil]

Use GetNearestPoint:
https://docs.unity3d.com/Packages/com.unity.splines@2.0/api/UnityEngine.Splines.SplineUtility.html#UnityEngine_Splines_SplineUtility_GetNearestPoint__1___0_Unity_Mathematics_float3_Unity_Mathematics_float3__System_Single__System_Int32_System_Int32_

It gives you t as an out variable