You forgot to tell what's your max. budget for that and whether you want a RME device for that.

You can add additional devices by using the ADAT port and connecting the devices by TOSLINK cables.

For A/D conversion you need only 1 cable, but to be able to clock sync the connected device it also needs ADAT inputs, so you need a 2nd TOSLINK cable, or you need to clock sync by using Word Clock.

AD/DA converter have usually between 8 - 32 ports in increments of 8.

So, you will need to get an 8 port AD/DA converter for this.
An alternative might be a Mic Preamp, where you can use the Mic inputs also as line inputs.

Easiest for handling might be a pure AD/DA converter.
The better ones support different reference levels, switchable per port.

What type of device do you want to connect? All balanced with TRS or XLR plugs and studio level (+4 dBu)?

If you want RME devices, I see the following options:

RME M-1610 Pro might be too expensive for you and makes no sense, as you have only one ADAT port on your UCX II.
This supports only 8ch at single speed. But the M-1610 Pro has 16 analog inputs (and 8 analog outputs).

The ADI-8 QS is an 8-port device, but also expensive with MADI, which you do not need. This design doesn't allow the configuration of different reference level per port, which might be a disadvantage for you.

An alternative might be an ADI-8 DS MK III from used market, this device: https://archiv.rme-audio.de/products/adi_8_ds_mk3.php. Also, here you can change the ref levels only for all the inputs and all the outputs, not per port if that should be needed.

So, if these devices are maybe too expensive (to get 4 analog inputs more), you need to look for 3rd party devices.

Don't expect RCA connectors on a RME AD/DA converter. ADI-2 DAC FS is an exception.
You need a converter cable for such an unbalanced connection.
So, you need to be able to connect balanced and unbalanced connections.

Regarding devices, it would need some time for research.
One ADAT port @96 kHz supports only 4 channels.
There are not RME AD/DA converters available with ADAT I/O and only 4 ports.

You could only connect an 8 port RME AD/DA converter, but then you can't use 50% of the ports.
And it requires a closer look whether consumer level is supported for the device with the RCA plugs.
All older RME designs/devices do not allow changing of reflevel per port.

Thanks again ramses! This is all very helpful.
As far as the RCA outputs on a couple of my source components, could I use a 1/4 TRS to RCA adapter with these RME devices?

I'll see what I can find on those models you mentioned. 

I assume I'd just connect one of the devices you listed to my UCX via the ADAT?