It's simple. That's because "визит" implies time in this context. "В последний визит" sounds like "in [time of] my last visit".
There is the close problem in English between "which/who" and "that".Also an other question that arises from reading a story of K.Паустовского :
"Он украл с нашего стола, что стоял около домика, кусок колбасы и, несмотря на наши крики, полез с ним на дерево."
Why что and not который (given that he refers to the word стол)
For example: I know a person who likes beer. I know a person that likes beer.
Although in Russian "что" (as "that") sounds a bit old fashioned.