Material distortion around edges

Hey, I'm applying a material to an object, but it's not wrapping around the edges naturally. I'm importing my objects from Photoshop.

In the attached example, the object on the left is a previously imported object where the material has wrapped successfully (not perfect) and the object on the right, is showing the problem on the new object.

I think it has something to do with the UV, but I have no idea what I'm doing here. I've used Object > Generate UV, which partially helped. But as you can see on the right object (rotated to demonstrate) the material fragments, where it should be seamless.