The 9-bolt top cover tranny used on the 6's was the same as the one used behind the 8 except for the fact that it had a lower 1st gear (makes up for the less powerful 6). It should be fully synchronized, so I would guess that there is something wrong with yours.
Lesson:
The 2.77 and 3.03 refer to the input shaft length, not a 'transmission code' or part number. The only 2.77s were used from 1960 to 1965. They have a 4-bolt top cover. In 1966, Ford replaced the 2.77 with a 3.03. However, this 3.03 looks just like the 2.77 except for the fact that the input shaft was longer. It still had a non-synchronized 1st gear and had a 4-bolt top cover. The real 3.03 as we all refer to it was used from 76 on and has a fully synchronized 1st gear and a 9-bolt top cover.
We all (myself included) refer to the '66 as a 2.77 because it is easier to recognize it as that with the 4-bolt top cover. That is why I always have to ask questions of people when they ask me questions about tranny swaps.
1) How many bolts hold the starter?
2) How many top cover bolts?
3 bolt starter and 4 bolt tranny cover
8 1/2" recessed-style flywheel, official 2.77 (60-65)
2 bolt starter and 4 bolt tranny cover
9" flat flywheel, pseudo 2.77 (66)
2 bolt starter and 9 bolt tranny cover
9" flat flywheel, official 3.03 (67 on)
when I get on a roll
sufficient confusion for all