C6 Automatic transmission, not shifting

We have an early model C6 transmission in our 68 cougar. Supposedly it was rebuilt. It is not shifting we can manually shift it from 1st to 2nd but it does not shift from 2nd to 3rd and when it’s in drive, it does not automatically shift at all.

Check the vacuum line going to the shift solenoid on the back of the transmission

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+1 to checking the vacuum to the modulator. Needs to be a full manifold vacuum source.

Not sure how long ago it was rebuilt or if at all, but a stuck governor valve can prevent up shifting automatically. You have easy access to it with the tail housing removed

Vacuum modulator gone bad or the vacuum line attached to it.

While the modulator (or just a disconnected vacuum hose!) will make it not want to shift until after 50 mph or so, and will bark the tires trying to put in Drive, low fluid (or too much fluid!) is also a common reason for no shifting.