How to fix old un-suppported basement wall in house that has had added basement?
This is probably just a good comment but I would highly suspect that the old wall is providing zero support.   Just because things are connected to it does not mean that it is supporting those things.   

Given that a new basement was poured there is a 99% chance that they pushed those walls up to provide load support to the entire house.   Since this was done in 1965 I am pretty sure I am correct or you would have a warped/bent floor on the edges.

The concrete that is malfunctioning on you is simply concrete breaking down because it is holding up itself.   Concrete is really bad at this.   That is why structures like parking garages require tons of rebar and meshing even though they are mainly concrete.   And even then concrete will crack and fall.   I am guessing yours has nothing internal supporting it.

My recommendation would be to fill in the void as well as possible with drainage rock and aggregate.   There are techniques you can use when adding the rock to pressurize the load for that small wall better.   But note that the load is only itself.   

The other note I would make is the most concerning part of your picture is the angle of the missing dirt.   It seems that the void clearance runs into your usable foundation.   I would add enough dirt to add a size ramp to offset any wicking water to the other direction (8-12 inches higher).