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I just became a member of the club a few months ago. Just before i turned 50, I bought a 2001 Pewter Metallic, Glass Top, Six speed. Although I drove a stick in Europe on vacation a few years back, I learned on this beautiful car. Unfortunately, it had a problem from the beginning.....seems to not like getting it in reverse. Sometimes it goes right in, other times I need to keep poking and waiting for the synros to get me there. And sometimes, with the clutch in, it will grind at times. Before I bring it in, anyone have any ideas what I'm in for? By the way, I don't down shift much but when I do, I rev match. And I'm sure I'm not the only one, but I rev match on the upshift as well.
I don't own a Corvette (yet). But as a mechanic I would say the shifter linkage is out of adjustment. Most likely someone didn't know what they were doing when installing a new clutch. I don't know if this car has a synchro for reverse, but I've never seen a problem with synchros in reverse gear. Another possibility is if clutch doesn't disengage fully, ie. leaky master or slave cylinder, throw out bearing. If all that works... transmission is the only anwer, but i doubt it. The most commonly failing syncros are 2 to 3rd, 1 to 2nd, 3 to 4th in that order. If the engineer did his job when designing the car, the weight of the flywheel should minimize the wear on the syncros on upshifts so revmatching is completely unecessery. Only way these fail if they are from an old 911 box or abuse. Abuse I mean driving in first gear, putting in the clutch letting the engine drop to idle, and the letting it out. That's abuse, tracking you're car is not.
Of course I don't know just how difficult yours is. What I do that helps is shove it into 3rd gear for a second first. Mine always drops into reverse easily after that.
That happened to me also it was popping out of first. The bolts that are attached to the torqe tube were loose so I applied some locktite to them and retightened the bolts.Those bolts could become loose over time from vibration.When the bolts become loose the alignment goes out.You have to take the front part of the dash off and the console off to get to it.There is instructions posted on how to remove those parts,not hard to do but there are some hidden screws behind the ash tray and temperature sensor.
I think if you read the owners manual, I think you will find that this is normal. I know I read it somewhere. And it is a safety thing that they want you to put it into another gear before going to reverse. When I park I leave mine in 3rd gear and the manual or people say you should leave it in reverse. I know when I go for reverse it does the same thing as you are stating. But if I go into 3rd then reverse there is no problem. Plus I changed to Amsoil tranny fluid and rear end oil and it's a lot smother. My .02