Just happened to me a week ago in WI, after the car sitting overnight with temps in the 20's. I turned the car on no problem, went to reverse, and found the shifter would only easily move forward and back between 3 and 4 which are the at rest gears (no side to side movement). I'd read about shift linkages disconnecting on the trans housing so I removed the intake and airbox housing to look at the trans and linkage (first time for this) and saw that both linkages appeared to be still attached. You can manually move the linkages like this and I was able to shift it in the back and forth way, but not side to side, but both linkages were definitely securely attached.
I took off the shifter boot and looked things over, everything still attached, but something was still binding and it felt like it was from the shifter itself. I sprayed down all the mechanical contact points/linkages (including the shifter base) with Liquid Wrench white lithium grease (doesn't freeze) and it freed up and now shifts better than it has in quite awhile. I don't know if this is really a permanent fix but it definitely solved what I thought was a bigger problem, and one that I'd encountered for the first time.
So I'd suggest taking off the shift knob and shifter boot and having a look at the shifter mechanism itself, something in there was binding for me and greasing it helped.