I had the same problem with the lyre spring on my 6497 clone... Flew off 3 or 4 times... but luckily found it each time. Good tip about the plastic bag @watchweasol! I may well go with that in future. Lyre springs are turning into my pet hate already. I broke a tiny one on the first practice watch I got to work on (FHF ST 96) barely touched the arm and snapped clear off... it was about the last thing I was doing after a full clean and reassembly too... I ordered a new one (not that it is economically worth it, the watch cost £1.70) but I didn't want to leave it unfinished now I had gone so far, with the particular lower shock setting the movement has it will give me practice removing and replacing the setting too.
The AS 2066 seems to be a quircky one took some photos of the setting positions.
As you see on the third picture the day is connected to the hour wheel, so every 24 hour it flips it.
But the date has a quick leaver in the second position.
When the lever spring isn't level or tight enough this causes the quick date set to not function.
Time and Date.