There is a nice result in geometry which says that if you take a region in the plane which has an area, a point P not in the plane and form a "tent" by joining every point on the boundary of the region to the P then the volume of the tent is
1/3 (the area of the region at the base) (the height of the tent)
Your oval-based cone fits this description,
and hence if you know the height you only need find the area of the oval.
If the oval is an ellipse then it has a major axis and a minor axis. Let R be half the length of the major axis and let r be half the length of the minor axis,
then the area of the ellipse is
Hence if the base of your cone is an ellipse then the volume is
1/3 r R h