Seems to be the space on the southeast corner of 8th & Congress, next door to the Stateside Theater.

Pieced this together via:

1) "[...] named the Avenue Hotel but locally known as the Stringer’s Hotel" (https://en.m.wikipedia.org/wiki/Swante_M._Swenson)

and

2) "The Jan. 18 announcement indicates that The Avenue’s name is an homage to the original structure built on the site in 1880, the Avenue Hotel." (https://www.bizjournals.com/austin/news/2019/01/18/31-story-...)