Not a SAS smock but a lot of other things so here are a few thoughts.
Use enough dye, I dyed a Judo uniform with just one box of dye and the brown turned into an almost salmon pink; I had to use two boxes of RIT just to dye a Tilley hat the deep brown shade I wanted, I also reused the dye a second time on the same hat.
If the thread is a different material you can end up highlighting the thread. If it was cotton with nylon thread, you could end up with the thread dyed a lighter shade than the jacket, this would make the thread stand out.
Keep moving the garment, if you don't you could end up with darker and lighter areas.
Use clean garments that are also free of stains or you could again end up with darker and lighter areas.
Use a long stick to stir the bucket (I use a bucket, the dye will stain your tub for a bit) and best to do it naked so you don't get dye splashed on your clothes (easy to do). Rinse your bucket right away but it still may dye. More concerning is to clean your toilet right away if you pour the dye water down it as that can stain for a while too. Be careful of splashing dyeing your floor or walls.
