A GeForce GTX 660 can drive four displays all by itself.
If your new displays have HDMI and DisplayPort video input ports then this is the ideal single card solution in your budget range without having to buy an active DisplayPort adapter.
Wow, I hadn't considered such a solution! 2 out of the 3 monitors have DisplayPort input ports. Thank you for reminding me to check the input ports, @ko888.
Tough call between upgrading to GeForce GTX 660
($229), or installing a cheap secondary card 6570
($50). I'm leaning towards the $50 option. I don't game that often these days anyhow. Thank you, @luciferano for that suggestion.