Persist on BIG-IP Session Cookie first else use use Server Cookie to select Pool

I'd apply a cookie insert persistence profile to the virtual server, using the default cookie name [BIGipServer].

Then in your iRule, first look for an LTM persistence cookie and pick the pool based on the cookie name.

If no persistence cookie exists, then inspect the server-set cookie. If it contains a new_site_users class member, select the new_site_pool.

If neither condition is true, select the original_site_pool.

Class:

class new_site_users {
  siteA
  siteB
  siteC
}

Rule:

when HTTP_REQUEST {
   first look for LTM persistence cookie & extract pool name
  set myPool [findstr [HTTP::cookie names] "BIGipServer" 11 " "]
  if {$myPool != "" }{
    pool $myPool
  } elseif {[HTTP::cookie exists "server_set_cookie"] && \
   [matchclass [HTTP::cookie "server_cookie"] contains $::new_site_users]}{
     pool new_site_pool
  } else {
     pool original_site_pool
  }
}