I have a 64 dart I use for winter driving.
It is a great car for winter and has saved my butt many time in the snow, thanks to the 7 1/4 shur-grip rear end.
The push button automatic will randomly stick in first for a bit while first driving.
When it finally shifts, it'll do second gear for a split second and end up in third.
I'm guessing a shift valve in the valve body or a sticky governor.
After the initial delayed shift, it won,t do it again that day.
I probably wont mess with it (especially this season) unless it gets worse.

A fluid & filter change is a great first step.
Be sure to use the correct filter... for a 60-65 "cable" 904... these have 2 oil pumps so look for 2 oil feed holes, in the correct filter.
(or you can add the second hole in the 66 & up filter)
DD

I'll have to be sure to get the right filter since I didn't know that they have a different filter.
A fluid and filter change helped since it was worse when I first started driving it 14 years ago.
It had 64k miles then but had been sitting and rusted out much of the underside before I bought it.
It now has 84k miles and idles so smooth that I have to look at the ammeter to be sure I don't accidentally grind the starter trying to restart it while it is running.

Having the later "1 hole" filter in the early 904 could be the cause of your late shift problems...
That is the first thing to check.
DD