As it stands right now... HOTin and NEUTRALin are connected
to the lower outlet's push in terminals, HOTout and NEUTRALout
are connected to the lower outlet's screw terminals, and SWITCHout
is also connected to the lower outlet's screw terminal.

I don't know if it's kosher to use the screws and push in terminals to
daisychain circuits... It does work so should be OK.

BUT you should only have ONE wire under a terminal screw. If you can't do
that, then you put a short wire under the screw and join it to the other
wires with a wirenut.