Thanks for your advice, everyone.
I tried LD_LIBRARY_PATH (because that seemed easiest):
# export LD_LIBRARY_PATH=/usr/lib
# make clean
# make OPENSSL=/usr/local/src/openssl-0.9.6c
I get the same error.
So it looks to me like the best route is going to be to download more current OPENSSL source code and link against that. Ug.