Is it possible to handle the ssl connection with IO::Socket::SSL directly but still have all the LWP::UserAgent functionality for my interaction with the server after the connection is established?
No, LWP wants to have full control over the socket and does not allow to create a connection using an already established socket. Apart from that, the following code works for me with LWP::UserAgent 6.05 and LWP::Protocol::https 6.04 (with the modification done in Ubuntu 14.04): 
use strict;
use warnings;
use LWP::UserAgent;

my $ua = LWP::UserAgent->new;
$ua->ssl_opts( SSL_ca_file => 'self-signed.pem');
# set verification name explicitly for this test because the
# URL does not contain the correct name
$ua->ssl_opts( SSL_verfifycn_name => 'foo.bar');
my $res = $ua->get('https://127.0.0.1:4433');
print $res->as_string;

[download]
In this case s_server is started with -WWW, that is: 
openssl s_server -cert self-signed.pem -key self-signed.pem -WWW

[download]
Which leaves the question about the version of LWP::UserAgent and LWP::Protocol::https you are using.

In reply to Re^3: HTTPS connection with LWP and self-signed certificate by noxxi
in thread HTTPS connection with LWP and self-signed certificate by Anonymous Monk

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