I upgraded my gen 8 dl380p with an extra 4 ports with hp 4port 331T 1gb network card. Esxi hypervisor does not see the new card. the esxi install is HP customised. I still have the original 311FLR hp on board card installed.
I assume that you know that the latest supported ESXi version for your host is/was ESXi 6.5?
Anyway, does the host itself recognize the additional network adapter (you may check that in iLO)?
If your card is exactly this one, this would be the supported driver and version for version 6.7U3.
In an SSH session on the ESXi Host, run the following commands.
To find out if we have installed the ntg3 driver and its version
# esxcli software vib list |grep ntg3
to get a list of the Nics with their details
# escli network nic list
To know if the driver is loaded and active
# esxcli system module list | grep ntg3 or
# esxcli system module list | grep 'Name|ntg3'