Yes, the shadow number is reduced by 1 (according to a CRD ruling back in 2002).
When the fellowship moves with Bill the Pony to an underground site, the Shadow number of that site is reduced before Bill the Pony is discarded.
Since I found this ruling a little counter-intuitive, I would like to attempt to explain the ruling. Hopefully someone will find this helpful.
The rules do not explicitly define when "at a site" actually is in effect. One might think that "at a site" is tied directly to which site the player marker is currently on (step 4). However, if we use the 2002 ruling as a guideline, it means that "at a site" does not occur while any of the movement actions are in progress (until after step 8).
<Fellowship or Regroup phase begins>
...
...
Free Peoples player announces they are moving.
------------------------The fellowship is "at" the old site until this point.----------------------
1) A Shadow player places the next site card, if needed.
2) Perform "When you move from..." actions.
3) Perform "When the fellowship moves..." actions.
4) Move your player marker to the next site.
5) Perform "When you move to..." actions.
6) Add twilight tokens equal to the new site's Shadow number.
7) Add 3 twilight tokens if the new site is in region 2; or 6 if it is in region 3.
8) Add 1 twilight token for each companion.
------------------------The fellowship is "at" the new site from this point on.----------------------
<Fellowship or Regroup phase ends>