Using external DAC with computer speakers

Most of the time, external audio solutions like the O2/ODAC combo, the Schiit stack, and the Audioengine D1 are meant for headphones only.  But I've been wondering what would happen in my case.

I mainly use my Logitech Z2300 for general use, gaming, and such, however there are times when I need to be quiet and I switch over to my AudioTechnica AD1000x's, which I believe are 40 Ohms.  I know that since my Z2300 are powered speakers, there should be no problem with just buying an ODAC and plugging the speakers directly into it, unless I'm not understanding how speaker amplification works.  The problem lies when I connect my headphones to the built in headphone jack on the speaker control pod.

Since I'm pretty sure this headphone jack is a passthrough and not in anyway connected to the amp in my speakers do I need to buy an amp in order to be able to power the headphones? Or can I get away with just having an ODAC/Schiit Modi in my setup?