Morning,
This is a common issue. eth0 is hard coded in /etc/sysconfig/network-scripts to a specific MAC address. Make sure your using ifconfig -a not just ifconfig... in this situation ifconfig will not display the new adapter because it does not have a specific config for it.
Do the following:
1.ifconfig -a
2. Get the mac address for eth1/eth_whatever it is
3. cd /etc/sysconfig/network-scripts
4. vi ifcfg-eth1
5. Make sure the max is correct and do your network settings are correct
6. Restart networking /etc/init.d/network restart
7. Your good to go.
If you want the new adapter to be eth0 then you have to remove the udev rules and reboot as suggested.
Please let me know if you have questions.
Thanks,