Unfortunately, the GetFramePointTangent method returned nothing in my scenario.
I was able to find a workaround by using the GetCurve method on the SplineSegment and then using GetBCurveParams5 on the resulting Curve to get the Curve's SplineParamData. The Spline's control points can then be taken by calling GetControlPoints on the SplineParamData object.
I then used the control point location closest to the Spline's endpoint to get an approximate of the desired tangent vector.
Curve segmentCurve = (Curve)currentSegment.GetCurve();
SplineParamData curveParams = (SplineParamData)segmentCurve.GetBCurveParams5(false, false, true, true);
bool statusTest = curveParams.GetControlPoints(out controlPoints);
double controlPointDoubles = (double)controlPoints;