Okay all, I have an answer!
JFK Visit – September 11-12, 1962:
Arrived Houston International Airport (now Hobby) at 6:15 PM
Stayed evening at Rice Hotel 909 Texas Ave, Houston, TX 77002 (Sept 11, 1962)
September 12:
Left Rice Hotel 9:34 AM
Motorcade travelled down Main St to University Dr, to Rice Stadium at 9:58 AM (route per El Paso news article, pasted below, and video shows the motorcade arriving off University)
Presidential address at Rice Stadium 10:15 AM to 10:30 AM
Travelled to the Rich Building, arriving 11:10 AM:
University to Main (per video and El Paso news article)
Main to Hermann Dr to San Jacinto to Highway 59 (Main per El Paso news article and video showing them headed toward downtown, educated guess on turning on San Jacinto) (Unsure if there was a way to enter 59 directly from Main, today you need to go San Jacinto to 59 feeder to get on 59 easily, but the elevated portion of 59 today was likely how it was in 1962 – it's been that way since as long as I can remember)
Highway 59 to Gulf Freeway (Gulf Freeway per El Paso news article)
Gulf Freeway to Reveille St / Route 35 (Gulf Freeway per El Paso news article, Reveille/35 is listed as a main route on the MSC map)
Reveille to Westover (educated guess, but Roundup notes they turned into the Westover entrance, and video clearly shows them arriving on Westover)
(Futher evidence the route information above is correct: The El Paso news article details as to time of arrivals and departures match extremely closely with the official JFK Visit minutes AND the article notes a "10 mile drive" from the stadium to the Rich Building – the route described above is exactly 10.7 per google maps, the author back then would have rounded and been just slightly less precise. Also, the article notes the speed of the motorcade would be judged by the crowds. If they took only streets, such as OST to Telephone, it would have taken a lot longer than 40 minutes to get to the Rich Building – Google maps says it's only 18 minutes by the route above on today's roads, but it makes sense they would have taken time going up Main to 59, and time on Reveille and Westover near the Rich Building. Even today, it is likely more like 25-30 minutes with some traffic to take the above route.).
Departed Rich Building for Houston International Airport 12:39 PM