I'm not sure what connections you have on the back of your particular card, but if you have display port, this is a good start.
a quick look at the spec of the 580 tells me it supports Displayport version1.4 with HDR.
display port version 1.2 brought in multistreaming, allowing you to split the port into 2/3/4 separate ports with a splitter box, or daisychain monitors together.
DP1.4 supports an actual data rate of about 26gigabits a second, and provided the display data for all the screens you want to connect to this one port fits into that data rate, then you should be fine. but it is still a limit, so if you're running some 1920x1080 screens, you'll be able to run a few of those, but if you're running 4k, then maybe only one or two.
other options are things like the matrox dual or triplehead-to-go, which takes a single output either VGA/DVI, or DP, and the box presents itself as an actual display - the capabilities it present to the graphics card then depend on the monitors attached to the triplehead-to-go. for example, connecting three 19" screens that can each display 1280x1024 to the matrox box, would look at the computer like a single display that runs 3840x1024. the max supported resolution of the matrox thing is 5760x1080 for 3x 1920x1080 screens, or 3840x1200, on 2x 1920x1200 screens
so your options are there, just depends what screen resolutions you are going to be running and your monitor layout.
where we were running 8x 19incher's at 1280x1024 we went to 3 or 4 30incher's at 2560x1600. now i'm seeing layouts for 2x 48incher's at 4k
otherwise yes, you can run multiple graphics cards in your machine. ideally both are from the same manufacturer AMD or Nvidia, and able to run from the same combined driver pack you've installed and even better that both are the same card/GPU. I do hear that with windows10, you're able to run both Nvidia and AMD cards in one machine much more reliably - but i wouldn't put it past either of them to pull a trick like "opponent's card detected! rebooting!".
i've run a pair of nvidia quadro M4000 8G's to drive a bottom row of three 30 inch screens (each 2560x1600) and a top row of three 27 inch screens (each 2560x1440).
i got annoyed that i couldn't SLI the quadro's in any other way than just having a big desktop, so pulled out one card and the 30 inchers.
just remember that your power consumption is going to double, and heat problems are going to more than double with two cards in the mix.