Carry session variables from https to http

I have just moved web hosts and have run in to a problem.

With my old host, on one of my client sites, when the user logs in under https, any session variable that was then set under https would then carry across to http. For example, a clientID that was set as a session variable would be visible under http.

Since I have moved sites, this doesn't seem to work anymore.

I don't know if this is a server setting...or if there is something I need to fix in my code?

Any help appreciated!