The new behaviour seems broken on my system.

$ PYTHON_VERSION=3.10.0

$ PKG_CONFIG_PATH="/home/chaz/.local/local/sqlite3/lib/pkgconfig:/home/chaz/.local/local/openssl3/lib64/pkgconfig"

$ LLVM_PROFDATA=/home/chaz/.local/local/clang+llvm/bin/llvm-profdata CC=clang CXX=clang++ ./configure --prefix=/home/chaz/.local/local/python-${PYTHON_VERSION} --enable-optimizations --enable-shared --enable-loadable-sqlite-extensions --with-openssl-rpath=auto

[..]

building '_ssl' extension
clang -pthread -fPIC -Wno-unused-result -Wsign-compare -Wunreachable-code -DNDEBUG -g -fwrapv -O3 -Wall -std=c99 -Wextra -Wno-unused-result -Wno-unused-parameter -Wno-missing-field-initializers -Wstrict-prototypes -Werror=implicit-function-declaration -fvisibility=hidden -fprofile-instr-use=code.profclangd -I./Include/internal -I/home/chaz/.local/local/openssl-3.0.0/include -I./Include -I. -I/usr/local/include -I/home/chaz/Downloads/Python-3.10.0/Include -I/home/chaz/Downloads/Python-3.10.0 -c /home/chaz/Downloads/Python-3.10.0/Modules/_ssl.c -o build/temp.linux-x86_64-3.10/home/chaz/Downloads/Python-3.10.0/Modules/_ssl.o
warning: no profile data available for file "_ssl.c" [-Wprofile-instr-unprofiled]
clang -pthread -shared build/temp.linux-x86_64-3.10/home/chaz/Downloads/Python-3.10.0/Modules/_ssl.o -L/home/chaz/.local/local/openssl-3.0.0/lib64 -L. -L/usr/local/lib -R/home/chaz/.local/local/openssl-3.0.0/lib64 -lssl -lcrypto -o build/lib.linux-x86_64-3.10/_ssl.cpython-310-x86_64-linux-gnu.so
clang -pthread -fPIC -Wno-unused-result -Wsign-compare -Wunreachable-code -DNDEBUG -g -fwrapv -O3 -Wall -std=c99 -Wextra -Wno-unused-result -Wno-unused-parameter -Wno-missing-field-initializers -Wstrict-prototypes -Werror=implicit-function-declaration -fvisibility=hidden -fprofile-instr-use=code.profclangd -I./Include/internal -I/home/chaz/.local/local/openssl-3.0.0/include -I./Include -I. -I/usr/local/include -I/home/chaz/Downloads/Python-3.10.0/Include -I/home/chaz/Downloads/Python-3.10.0 -c /home/chaz/Downloads/Python-3.10.0/Modules/_hashopenssl.c -o build/temp.linux-x86_64-3.10/home/chaz/Downloads/Python-3.10.0/Modules/_hashopenssl.o
warning: no profile data available for file "_hashopenssl.c" [-Wprofile-instr-unprofiled]
clang -pthread -shared build/temp.linux-x86_64-3.10/home/chaz/Downloads/Python-3.10.0/Modules/_hashopenssl.o -L/home/chaz/.local/local/openssl-3.0.0/lib64 -L. -L/usr/local/lib -R/home/chaz/.local/local/openssl-3.0.0/lib64 -lssl -lcrypto -o build/lib.linux-x86_64-3.10/_hashlib.cpython-310-x86_64-linux-gnu.so
*** WARNING: renaming "_ssl" since importing it failed: libssl.so.3: cannot open shared object file: No such file or directory
*** WARNING: renaming "_hashlib" since importing it failed: libssl.so.3: cannot open shared object file: No such file or directory

$ ldd /home/chaz/Downloads/Python-3.10.0/build/lib.linux-x86_64-3.10/_ssl.cpython-310-x86_64-linux-gnu_failed.so
        linux-vdso.so.1 =>  (0x00007fffc99f4000)
        libssl.so.3 => not found
        libcrypto.so.3 => not found
        libpthread.so.0 => /lib64/libpthread.so.0 (0x00007faa70802000)
        libc.so.6 => /lib64/libc.so.6 (0x00007faa70434000)
        /lib64/ld-linux-x86-64.so.2 (0x00007faa70a1e000)

$ readelf -d /home/chaz/Downloads/Python-3.10.0/build/lib.linux-x86_64-3.10/_ssl.cpython-310-x86_64-linux-gnu_failed.so | grep PATH

It appears that RPATH/RUNPATH is not getting set.