You can connect your monitor to a Thunderbolt port using a DisplayPort to Mini DisplayPort cable like this one; other lengths are available if 3 Feet is not long enough for your purposes.
I connected my LG 25UM65 (2560x1080 native resolution) to my MacBookPro11,3 (mid 2014 15" Retina) three different ways, HDMI to HDMI, Thunderbolt to DisplayPort, and Thunderbolt to HDMI. The Thunderbolt to HDMI connection is not useful (the maximum resolution it offers is 1920x1080), but the other two connections both work at 2560x1080@60Hz, and the visual quality is comparable across the two connections (but both look blotchy compared to the Retina display). A UHD display (3840x2160) is a better choice than 2560x1080 to maximize the display quality of Thunderbolt 2 (DisplayPort 1.2).
It's possible that a better HDMI cable will improve the display quality (the one I'm using is labeled HDMI HIGH SPEED). You can determine (and change) the resolution and refresh rate from System Preferences (Display) by choosing
Scaled rather than
Default for display (using the Alt key while choosing
Scaled can provide more choices on some displays). A 2015 Mac's HDMI port is capable of driving 2560x1080@60 Hz, provided the display has a suitable HDMI input (HDMI 1.3 is sufficient, but HDMI 1.2 probably isn't). If your display has multiple HDMI inputs, you might try all of them to see if that makes a difference.