The shift to the Jackson comes about just before the European leg of the Lick it up tour. It's not shown in the tour book photos shot at SIR studios but instead makes its "proper" debut in the video for All Hell's Breaking Loose and can be seen briefly in the footage from Madrid 83.10.14. (And once the American tour book is printed there is a shot of the new Axe for good measure.)

The Jackson Axe continued the new infatuation with EMG pickups, this one located closer to the neck than it had been on the Kramer. The headstock was the standard Jackson "hockey stick" with all four tuners on one side. The headstock and the space between the neck and the bridge had tortoise shell overlays and the bridge was once again hidden from view by a bridge cover.

One curious detail is that the picture from London 83.10.23 appears to show strips of tape making up the black parts of the Axe body. It's harder to see in other photos (none I've found show it quite as well as this one but the one to the left is also decent) but thankfully there is a horrible closeup of Gene's tounge from the 1984 Tourbook (far right below) that shows a detail of the Jackson Axe and the tape on the body. This implies that the taped-up part was probably finished silver originally.
Check I love it loud and I still love you from Madrid to catch a few glimpses of the Axe.
The Jackson Axe is mercifully kept in this near-original condition for at least the two following tours; there are no noticeable alterations. The most high-profile appearance during 1984-1986 is in Animalize Live Uncensored where it gets taken for a spin for Gene's solo spot.

Photos showing the Axe during the Asylum tour, where it did look a bit out of place against the exploding layers of sequined fabric and garish make-up, are a bit lacking but thankfully there is video evidence.
Then things get complicated...
 
 
 
In 1987 Gene turns up with what looks like a slightly altered version of the Jackson. There's a high-gloss, mirror-like finish on the body and headstock, added rivules on the body near the pickup, and, true to form, a hole where there should be (or had been) a tone knob. Apart from those changes it appears identical to the Jackson right down to the black tuning pegs. But, during the Convention Tour of 1995 this bass was part of the museum and was labeled as a prototype Gene Simmons Axe (see pic and video below). The very obvious absence of an Axe during 1990-95, with the possible exception below, might suggest that Gene was in fact trying out various prototypes in preparation of going into the Axe business for himself. Or maybe the rather strange decision to use the Axe for a song like Reason to live during the Crazy Nights tour drove home the point that the Axe was a somewhat odd fit for the image at the time. Whatever the reason for the lack of an Axe, this particular Axe makes its last appearance during the European leg of the Crazy Nights tour, shows up for the convention and then disappears; it doesn't even feature in the auction (which a lot of other Axes do).
Overall there were some factual errors regarding the instruments during the Convention and the tour book was laughably incorrect. It showed a mirror image of the #1 Spector and erroneously dubbed it a "Spectre" bass, even claiming it was the first "Spectre" ever made. The caption next to this Axe bass noted that it was a limited-edition built by Kramer.
I think it's fair to say that Gene's techs have made quite imaginative use of simple black tape throughout the years. It also speaks volumes of Gene's "working-bass" concept: as long as it works and looks good enough, it will do.