We played this piece this summer at the Blossom Music Center, so the measures I took to make the soft playing as comfortable as possible are fresh in my mind.
The challenges start right at the beginning with the famous solo. Along with the Rite of Spring, this is one of just a few solos for the bassoon that open a major piece of standard repertoire. Like the Rite, it is in a difficult register for solo bassoon writing.
There are many solutions to helping the first E start softly, securely and down to pitch. Below is the fingering I use -- notice that the register lock is on (see the strike through on the whisper key) to allow for safe passage to the F# in the solo.
On my bassoon I have a lever that partially closes the low B when the low Bb is depressed. You can make one of these for yourself by cutting a small strip from the backing of a notepad (like cardboard only thinner) and placing the strip in the linkage between the top end of the low B key and the arm from the B pad cup that overlaps it. This will close the B pad partially. Try different thicknesses. A match will also work.
Just remember to take it out when finished! Also, be sure to lock the whisper key when playing this solo, so it won't pop open the bocal vent!
At the top of the second page of movement one and at the end of movement two there are some soft low A's to play. If you need to cover your sound, try either of these fingerings:
Low Bb and Low D key are added to 3 of the 4 notes. The low F# may balk with this fingering combination, so I use the usual muffled F# fingering for that note, going back to the Low Bb/Low D combination for the Low D. Note, once again, the whisper key lock is on for safety!
If that's not soft enough, you can play with a mute in the bell.
This fingering may help, but be sure it doesn't make the C# too flat in pitch.