... You may also find there are opportunities at your local level as well. ...
OK, thinking locally:
USFS: Usually, if you go to the local district ranger station and ask about volunteering, you will get a positive response. Depends somewhat on the district ranger's philosophy, though. That's what I have done here in the Lincoln. The duties haven't been glamorous (staffing an information station during a forest closure, picking up trash and trail building), but it's work that needed doing. I hope to do more when I retire.
The other organization I know of that is active in the Lincoln NF (and elsewhere) is
Rails to Trails
And in New Mexico in general, there is
New Mexico Volunteers for the Outdoors. Don't know much about them other than they exist.
Then, there is the
New Mexico Wilderness Alliance. This is considered a radical group by the ranchers in these parts, ever since they started advocating to make Otero Mesa a National Monument.
They also advocate for roadless areas, which upsets the local off-road enthusiasts. I don't want to stray into politics here, but the group does have volunteer activities. I know there are some New Mexico off-road organizations, too (to present the other side of the coin), but don't know who to recommend. Maybe Barko knows?