Should I opt for a sliding or chin augmentation? Can a chin augmentation give me a small vertical projection?

I think the crux of your question is how much more projection do you need? If more than a couple millimeters, often the best way to widen is with an implant, especially if you already had genioplasty. If your projection forward is where you want it, then I would re-do the genioplasty, keeping it in the same position antero-posterior, but making a vertical cut in the genioplasty and widening it (either with a graft-I recommend non-bovine material or a wedge that you can often get from the same area). It is not easy to do, but that is why the projection matters so much. And you can do a small vertical projection with an implant, I would just do it from the outside, under your chin. If you do it from the inside, especially when you have had a genioplasty is a setup for the implant to ride up toward your lip.

In cases like this, I generally recommend a chin implant be placed to finalize the aesthetics. The implant can make the chin wider and/or add more projection as needed. I prefer to place the implant through a submental incision though I sometimes placed them through an intraoral incision.