I don't see how multiplexing is going to work in a DIMM slot. How is
SPD going to work when there are 2 modules accessed in the same mobo
slot? I think those old slot doubler adapter cards you remember were
for SIMMs, not for DIMMs.

https://en.wikipedia.org/wiki/DIMM
"Since Intel's Pentium, many processors have a 64-bit bus width,
requiring SIMMs installed in matched pairs in order to populate the data
bus. The processor would then access the two SIMMs in parallel. DIMMs
were introduced to eliminate this disadvantage."

I'd say to sell off the old memory (after testing with memtest86) to
offset part of the cost of getting bigger memory modules in your current
mobo. As old technology gets more sparse, like DDR3 (240-pin slot), it
commands a price premium (becomes more rare, longer shelf storage).
DDR4 (288-pin slot) is the currently saleable product, and is cheaper at
the same capacity.

https://www.newegg.com/p/pl?N=100007...27642%2080 00
32 GB (2 X 16 GB) DDR3 1600 (PC3L 12800)
$271 (sold by Newegg, not some 3rd party using a storefront there)

https://www.newegg.com/p/pl?N=100007...27642%2080 00
32 GB (2 x 16 GB) DDR4
$144 - $169 (did not include the CyberMonday deals)

For the price difference of $102 to $127, you could get a new mobo. The
problem is whether the old CPU works in the new mobo. I think the old
AMD Athlon x4 845 Quad CPU used an FM2+ socket, but I found one using
the AM4 socket.

https://www.newegg.com/p/pl?N=100007625%20600474773
Those have FM2+ CPU sockets. However, Newegg only has models with 2
DIMM slots, and they're DDR3, so you wouldn't be improving your
situation.

https://www.newegg.com/p/pl?N=100007...&Order=PRI CE
If your CPU uses an AM4 socket, lots more choices, and some at less than
the cost in the difference between the old DDR3 and new DDR4 price (and
some mobos also have 4 DIMM slots giving you later expandability).

You'll have to surrender to continue using your old computer for longer
until you can afford a better one (like when you get your tax refund),
or commit to surgery on your old box by replacing DDR3 with DDR4 which
means a new mobo, too. However, you'll probably want to bite the bullet
and get a better CPU, too.