how would you represent an image with a 60° span without just leaving most of it black?

You can't do it without leaving most of it black. With a 60ºx60º image, you've only covered 1/18th of the 360ºx180º view, after all.

Cardboard, being first and foremost a VR viewer, requires equirectangular input to represent the entire VR environment, so whether or not you are interested in creating 360ºx180º panoramas, that's what it's expecting: either a single equirectangular for mono view, or two stacked equirectangulars (left on top of right) for stereo view. But you have to have equirectangulars because most users of Cardboard are gonna want to turn their head to see what's to the sides and behind them.

If you want to use only a single 60º-coverage image, you're going to have to fill out the rest of the "space", somehow. The simplest way would probably be to just load your image into Hugin, set your projection to be Equirectangular, and stitch/output the panorama. The steps in my 2014 OSX version of Hugin are:

1. Open Hugin.

2. Select Interface → Advanced, because the default Simple interface won't let you stitch a single image.

3. Click the Add Images... button, and load the left view.

4. Make sure the Lens Type is Rectilinear, and if the EXIF didn't pull in focal length and crop factor, guesstimate your field of view.

5. Click OK.

6. Click on the GL preview button it the toolbar.

7. Select the Projection tab.

8. Make sure the Field of View is set to 360 x 180, and that the Equirectangular projection is selected.

9. Close the GL preview window.

10. Click the Stitcher tab.

11. Set your output format and size, and click the Stitch! button to make your equirectangular.

12. Repeat all this for the right eye image.

13. Open both equirectangulars in the Gimp, and make a square image with the left-eye equirectangular above the right-eye equirectangular (equirectangulars are always 2x1 aspect ratio).

But you're basically going to turn this:

into this:

If you're not starting with equirectangulars.

Shooting and stitching a high-quality 360ºx180º is both equipment and labor intensive, so within your limitations of a) only using an Android phone camera and b) not wanting to spend a large amount of time post-processing, the Google apps (Cardboard Camera, Street View) that do Photospheres are probably your only way to do this, and you won't be able to achieve high resolution/quality, because you can't control the stitch, and you won't have the best optics or control for the job, not to mention a way to control rotating around the camera's no-parallax point, if you're shooting in a smaller space.

Typically, people who shoot these kinds of images do so with a fisheye lens on a dSLR, with a panorama head, and stitch with packages like PTGui or Hugin. The Gimp or Photoshop alone can't do this particular job. And this type of shooting/stitching isn't particularly easy to master. (See How are virtual tour photos taken?). And then you'd have to find some way to duplicate Cardboard Camera's computational ability to create stereo images from a single image, much as 2D-shot movies get 3D-ized.

Everybody wants high quality for free and little-to-zero effort, but it's rarely a possibility.