Procedure to upgrade Net::SSLeay:

1. Find versions of Net::SSLeay and OpenSSL that support perl-5.6 and TLS1.2 and each other. Use the latest version of OpenSSL that you can.
2. Install upgraded OpenSSL where the build procedure for Net::SSLeay can find it.
3. Unpack a new Net::SSLeay and run perl Makefile.PL && make && make test. If tests fail, try another nearby version of Net::SSLeay until you find one for which the tests pass. After your new Net::SSLeay has passed its tests, use make install to install it.
4. Perl programs should now be able to use TLS1.2.

Then get a migration path for moving the entire program to a modern perl interpreter planned, because there is probably no combination of Net::SSLeay and OpenSSL that provides TLS1.3 support in a perl-5.6 installation.