Since posting my question I tried using HTTP::Tiny under 5.42.0 and it said:

IO::Socket::SSL 1.968 or later must be installed for https support
Net::SSLeay 1.49 or later must be installed for https support

When I tried to install IO::Socket::SSL it tried to install Net::SSLeay and that failed 3 tests but the way it failed and the MANY failures shown on fast2-matrix.cpantesters.org/?dist=Net-SSLeay+1.94 suggest to me that there are fatal bugs in the test suite for Net::SSLeay. Check this out:

t/local/32_x509_get_cert_info.t ............. 1/746 
#   Failed test 'X509V3_EXT_print nid=103    extended-cert.cert.pem:4'
#   at t/local/32_x509_get_cert_info.t line 273.
#          got: 'Full Name:
#   URI:http://intermediate-ca.net-ssleay.example/crl1.crl
# 
# Full Name:
#   URI:http://intermediate-ca.net-ssleay.example/crl2.crl
# '
#     expected: 'Full Name:
#   URI:http://intermediate-ca.net-ssleay.example/crl1.crl
# Full Name:
#   URI:http://intermediate-ca.net-ssleay.example/crl2.crl'

#   Failed test 'X509V3_EXT_print nid=86    extended-cert.cert.pem:6'
#   at t/local/32_x509_get_cert_info.t line 273.
#          got: 'email:intermediate-ca@net-ssleay.example, URI:http://
+intermediate-ca.net-ssleay.example, DNS:intermediate-ca.net-ssleay.ex
+ample, Registered ID:1.2.0.0, IP Address:192.168.0.1, IP Address:FD25
+:F814:AFB5:9873:0:0:0:1, othername: emailAddress:ica@net-ssleay.examp
+le'
#     expected: 'email:intermediate-ca@net-ssleay.example, URI:http://
+intermediate-ca.net-ssleay.example, DNS:intermediate-ca.net-ssleay.ex
+ample, Registered ID:1.2.0.0, IP Address:192.168.0.1, IP Address:FD25
+:F814:AFB5:9873:0:0:0:1, othername: emailAddress::ica@net-ssleay.exam
+ple'

#   Failed test 'X509V3_EXT_print nid=85    extended-cert.cert.pem:8'
#   at t/local/32_x509_get_cert_info.t line 273.
#          got: 'email:john.doe@net-ssleay.example, URI:http://johndoe
+.net-ssleay.example, DNS:johndoe.net-ssleay.example, Registered ID:1.
+2.3.4, IP Address:192.168.0.2, IP Address:FD25:F814:AFB5:9873:0:0:0:2
+, othername: emailAddress:jd@net-ssleay.example'
#     expected: 'email:john.doe@net-ssleay.example, URI:http://johndoe
+.net-ssleay.example, DNS:johndoe.net-ssleay.example, Registered ID:1.
+2.3.4, IP Address:192.168.0.2, IP Address:FD25:F814:AFB5:9873:0:0:0:2
+, othername: emailAddress::jd@net-ssleay.example'
# Looks like you failed 3 tests of 746.
[download]

Notice the difference between "got" and "expected" are very subtle and minor.

cpanm -v --force Net::SSLeay got the module installed so IO::Socket::SSL would install and now HTTP::Tiny works.