Just a reminder: the library may not encrypt their staff connection, but they probably still don't want you regarding it as a public access point. Rather than encrypting it, they may have put in firewall settings to exclude the public computers specifically, to keep their staff connection faster. In most places, it's still illegal to log onto an unsecured network without the administrator's permission, regardless of how poor their security is, the same way it's illegal for you to walk into a stranger's house just because their door is open.
I believe in network configuration or add a new connection, you can tick a box that says "Force this connection." You have to dig for it, if it's there, and you probably shouldn't dig for it on a computer you don't own or have sysadmin/root permissions for.
Even though it's selfish of the library to keep two channels for staff and only one for everyone else, they could file charges if you try to change that.