I should have noticed this before, but due to the CPS sampling design (most notably the 4-8-4 pattern), it is actually impossible to link the September supplement to the March ASEC sample since they are more than 4 months apart. So if you tried the method described in the working paper, you’d come up with zero matches. One possible alternative to match income information for people in the September supplement would be to grab the EARNWEEK variable. The limitation here, however, is EARNWEEK (and other Earner Study variables) are only asked of households in their 4th or 8th month in sample (see more info on this here). Although this could be a worthwhile alternative, it may be tricky to interpret because the information included in the Earner Study is collected at different points in time for different people.
I’m sorry this isn’t an easy solution to your question.