Your first statement above is correct and matches the murder arrest affidavit. The explanation for how this happened is more complex and is NOT in that affidavit. In my opinion, it is almost certainly due to him picking up the blue tarp (again) at around 9 am in the shopping area, along S Main St that was searched extensively by police earlier in this investigation. Here, it appears that police know BS twice visited that parking lot.
The first visit was around midnight on Sunday, 10/6, where "midnight" is defined as the end of day (as it normally is). In that first visit, BS seemingly dropped off the blue tarp with the large bulky item wrapped and secured in it. In this case, that was about a 30-min drive from near the wooded/brushy area, so the round-trip time was about 1 hour, as observed by his neighbor and reported in the murder arrest affidavit.
The second visit was about 9 hours later at about 9 am, 10/7. In that second visit, BS seemingly picked up the blue tarp with the large bulky item wrapped and secured in it (from where he left it 9 hours before). Here, a good guess is in one of the two large green Waste Management dumpsters in the mobile home park adjacent to that parking lot. But there are other good guesses like that small green Waste Management dumpster by the Alamo Hospice or under the many trees north of that parking lot that separate it from private property. Police must know exactly where that was with business surveillance cameras and the black box from the truck. From there, I'd guess he drove his black truck on north on S Main St, west on W Bandera Rd (which turns into State HWY 46 W), and turned left (in the left turn lane) into the Home Depot / Whataburger parking lot. That drive took just 5 min.
Regarding the Kendall County Solid Waste Site, he just discarded of the 3 white trash bags there according to the affidavit, probably in a blue Republic Services dumpster on the north end of the site. The south end of this site is for recycling. The north end of this site is for regular trash. Here, I'd guess he used the north end. He may have used this trip to decide if it was a good place to leave "the large bulky item wrapped and secured in a blue tarp". My guess is that he decided it wasn't a good place for that. Instead, my guess is he decided to gas up, buy water (to use with the Quikrete), and head about 30-min-drive further west, near the ranch, to do that critical disposal.