Take a byte out of it!
Jelly, 6 bytes

BḄ-8ƤṀ


A monadic link taking a number and returning a number.

Try it online!

How?

Uses a nice quick, Ƥ, developed by miles...

BḄ-8ƤṀ - Link: number
B      - convert to a binary list
    Ƥ  - for loop over some slices to be determined...
  -8   - this is a negative nilad, therefore: use overlapping outfixes of length 8
       -   (exactly what the specification asks us to inspect)
 Ḅ     -   convert from a binary list to an integer (vectorises)
     Ṁ - maximum