Push rods coming loose

I am thinking the sticking valve when cold is most likely the culprit.

I think a dose of Marvel mystery oil is in line.
Or maybe a quart of synthetic oil.

The car runs fine but it is smoking a little more now than before.

Thanks for your help.

David

Yep.
Even if the lifters did bleed down, they are still spring loaded and wouldn't let the pushrods come out.

This can only be two possibilities.
1. the rocker shaft bolts are backing out. (this would be really obvious)
2. the lifter is returning to the down position faster than the valves are returning to closed position. (allowing a big enough gap for the rod to jump out)

Mystery oil might do it, or a good dose of lead additive in the engine oil.
Either one will require you run it for awhile before any change would be noticed.
I have heard that Seafoam is pretty good stuff but have never tried it myself.