You are brave to tackle the jacket! I haven't done anything that I had to unsew! And I rarely do someone's own apparel. They order the apparel and the design, then I do the work. The good thing about this is that if I "screw up" I haven't ruined their favorite piece! I can order another piece and they'll never know I goofed the first time!
There is an easy way to make sure your design is straight in the hoop. I see in your photo that there is both a vertical and horizontal seam on your jacket. That is a great thing to use to verify that you are hooped straight. Line up the needle (or placement indicator) near the top of the seam. Then move your hoop up and watch the needle or indicator follow the seam, If is does - great! If it doesn't you can re-hoop to straighten, then repeat the procedure to check again. You can do this either vertically or horizontally, or both.
Of course, if the seam isn't straight, neither will be your design. But I think visually, sometimes it's better to match the seam (or other visual reference) than the true straight. I always try to visualize what the finished product will look like. I just did a v-neck tee where the point of the v wasn't exactly in the center. It wasn't off by much, but enough that if I had correctly centered the design it would have looked "off"! It looked great when done, and unless someone measures down to the 1/8th inch they wouldn't know! And on a moving body it looks perfectly lined up!
If you start doing a lot of jackets, or other lined items, there are blanks you can get that are designed just for embellishing. They have "ports", entry points, sometimes opened by an invisible zipper, so you can access traditional embroidery points!
Happy Stitching - SunRai Pat