Hi Merill,
SSL session reuse should be independent of the TCP connections as clients can resume a session on a new TCP connection.
I'm not sure why you'd see this with a default client SSL profile as LTM should cache sessions. Has the default client SSL profile been modified? What is the Cache Size set to?
You can test whether LTM is allowing reuse using opessl:
from: http://royontechnology.blogspot.com/2008/01/how-to-find-out-if-server-supports-ssl.html
openssl s_client -reconnect -state -prexit -connect ServerURL
Assuming the openssl test shows support for session resumption, maybe there's an error happening on the initial session which triggers LTM to remove the session from its cache? It's a stab in the dark.
It would probably be fastest to open a support case on this to get help troubleshooting the issue.
Aaron