Pictures?

That's the way houses used to be framed. I guess too many failed (sometimes the joist pulls away from the ledger) so the joist hanger was one solution. I see nothing wrong with your approach. You'll have to make sure the added 2x doesn't split as the nails will be close to the ends. You might even look at something other than a joist hanger, maybe an angle with holes located so you could use 3" screws to run through the new 2x into existing beam.

Here are some pictures, the header is about a 10 foot span, two 2x8's with a 2x2 ledger, also no hangers at each end. The joists that rest on the ledger run about 11feet to the sill plate atop concrete basement walls. They are in good shape, but this 10 foot section has sagged almost 3/4", at the stair case here, there is a staircase above it on the first floor and the ends of this header section in the basement are supporting a section of the 2nd floor with no real support down at the basement.

I have some lally columns so I wanted to raise it up a little at a time over a week or two and leave the columns in place at the ends of the span and that's where I started to think if I did that the joists here may be under enough stress to cause an issue since they are notched on the ledger.

I might also see if I can slip in a triple joist hanger on the right and left ends of the header to affix them (with ledger) to the joists that span to to main steel beam. Don't think I can get one in place where the stair case starts but maybe it doesn't matter as much if a column is under it.

A joist resting on a ledger is still allowed and notated as such in the building code.