I am trying to figure out the genre of this song so I can find more like it, but don't know how:
Specifically, the section from 1:07 till 1:57
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I don't know if that specific subgenre of rock has a single name. There are a good number of examples from the early-to-mid 1970s that used the horns and organ in a similar way with a rock band.
Are you looking specifically for instrumental-only music? Does it have to be in waltz time like that passage?
If you're looking for more like that passage and you don't mind some vocals, I would suggest starting with The Band though they didn't always use the horn section. "Ophelia" is an example where they do.
There are many other examples from that era, and many artists came later that were deeply influenced by them. A totally-random example from about a decade ago is Sam Winch, "OneTwoThree".
This is good old 70s soul, most reminiscent to me of Al Green.
Soul was a primarily Black American subgenre of R&B, innovated in the 60s as a secular version of gospel music, by figures such as Ray Charles, Sam Cooke and Aretha Franklin. The style also had a big influence on the white rock music of the times, leading to huge hits such as "Let it Be" (the Beatles), "Bridge over Troubled Waters" (Simon & Garfunkel) and "Whiter Shade of Pale" (Procol Harum).
Soul really came into its own in the 70s, led by singers such as Al Green and Isaac Hayes, who slowed down the tempo in order to focus on smoky, sexy ballads, anchored by organs, clean guitars, and gospel style background vocals.
An updated version of the soul sound was revived in the late 90s and early 2000s as "Neo-Soul." (Jill Scott, Angie Stone, D'Angelo). Both soul and neo-soul are popular again today among retro bands such as the Black Pumas.
Here's another recent tribute to that classic 70s soul sound.