I always shoot in center point focus. And for moving horses you want to use Servo mode.
It might help if you posted an image so we can evaluate you are talking about. Are you truly missing the focus plane or do you have a narrow depth of field that is giving the impression of oof?
If the former, it is just going to take some practice nailing the focus. With H/J if you are following the horse as it approaches the jump and your center point gets off the horse, when you fire the shutter you may have the focus locked on something in the background. A trick that I have learned is I keep the focus point on the point of the horse's shoulder. That is going to be at about the same plane as the rider's face. Depending on what aperture you are at you are going to have some depth of field in focus in front of and behind the rider. (More about that). Another trick is to preset your focus. If you are working the same jump, put your focus in "one shot". Focus on the middle of the top rail of the jump. (Or manually focus.) Now turn your autofocus off. And shoot away. Of course, check it every so often to make sure that you haven't accidentally bumped your focus.
If the latter, you may not have enough depth of field. You may have the rider in focus but if too narrow a dof, the horse's face may be out of focus. For heaven's sake, get your camera out of Sports mode and take control of it. Shoot in manual mode, set it at high enough shutter speed to stop action (your 1/1250th is good), then up the ISO enough to get stopped down enough to get an acceptable dof. Shooting outdoors on a sunny day I am typically at 1/1500 - 2000, f/5.6 and ISO of 400. If you don't understand this stuff then get something like Bryan Peterson's Understanding Exposure.. At 5.6, depending on the distance I am to the horse, I will have most everything in focus from the nose of the horse to it's tail. But still getting a nice oof background.
Lastly, the unfortunate reality is that some cameras and lenses are faster focusing than others. And the continuation of this reality is that fast focusing cameras and lenses are expensive. Doing this for a living I shoot with with 1D series camera's and mostly "L" lenses. You may have to just live with missing a few (or invest in faster focusing equipment).
HTH