I checked a little bit further and found the same brokenness in both SP-5.24.2 and SP-5.24.3.
SP-5.24.3 (still broken) had the following versions installed: 
D:\>perl -le "print $];"
5.024003

D:\>\b\pmver WWW::Mechanize
1.86

D:\>\b\pmver Crypt::SSLeay
0.72

D:\>\b\pmver Net::SSLeay
1.81

D:\>\b\pmver IO::Socket::SSL
2.051

D:\>\b\pmver Net::SSL
2.86

D:\>\b\pmver LWP::Protocol::https
6.07

[download]
Its openssl version was 1.0.2l

UPDATE:
Upgrading IO::Socket::SSL to version 2.056 (on SP-5.24.3) did not fix the problem - so it would seem that the issue lies NOT with any of those perl modules.
And I couldn't see any flags being waved in 5.24.4 perldelta, so I guess the problem is with either the openssl version, or some other module.

Cheers,
Rob

In reply to Re^2: WWW::Mechanize cannot connect to HTTPS site by syphilis
in thread WWW::Mechanize cannot connect to HTTPS site by Marshall

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