It does sound like a VLAN Trunking issue. You will need to make sure the ports connecting the new switch and the old switch are configured as trunk ports, and are passing all of the VLANs. You will also need to make sure the ports on the new switch facing the host are configured as trunk ports, and allow the appropriate VLANs.
Since you can connect the host to the old switch and it works that does a good job of eliminating the host vSwitch configuration.
