Data::Dumper is a good tool to visualise complex data structures, not to access them. In this case, you could access the JSESSIONID directly (since domain and path seem to be constant):

print $jar->{COOKIES}{'example.com'}{'/'}{'JSESSIONID'}[1], "\n";
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But there is a better way. Since $mech->cookies() returns an HTTP::Cookies object, you should use its methods to access the cookie. The scan() method should accomplish what you want.

# ... as before, set $jar = ...

sub callback {
    #  0  version
    #  1  key
    #  2  val
    #  3  path
    #  4  domain
    #  5  port
    #  6  path_spec
    #  7  secure
    #  8  expires
    #  9  discard
    # 10  hash

    #-- add more comparisons to narrow the search result
    print "Cookie: $_[2]\n"  if $_[2] =~ /^8430/;
}

$jar->scan( \&callback );
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See also perldsc, HTTP::Cookies, and maybe Data::Diver.