Can we characterize the following kinds of plane coverings? (Open-ended, but provide some description more "useful" than the constraints given.) For a more answerable question, is there an effective method for continuing any initial finite placement of line segments (or proving it impossible)? Do some initial placements force periodic coverings?

We must cover the plane with line segments such that each is parallel to two other line segments on opposite sides at unit distance. Each line segment must touch one or two others at its endpoints, and the angle between them must be 120 degrees. Moreover, the "paths" between segments must not have any branching.

Several obvious coverings satisfy the above. E.g., a hexagonal grid with unit spacing works, as does a banded pattern of "zig-zag" lines, or a series of concentric hexagons going out to infinity. There are less obvious coverings, such as:
     

In case the above verbal description is inadequate, I think it is equivalent to describe the problem in terms of requirements on local neighborhoods of individual points, illustrated in the following figure:
Local neighborhoods may be rotated as needed.

(1) Every point is on a line segment with parallel segments at unit distance.

(2) Segments can only terminate if they meet one or two other segments at a 120 degree angle.

(3) The "paths" between segments cannot branch. Specifically, the neighborhood shown with a red dotted boundary cannot appear anywhere in the covering.