How does one correctly use a "-" outside of hyphenation - if at all?

Solution 1:

That's not a hyphen, it's an (em) dash:

A dash is a punctuation mark. It is similar in appearance to a hyphen, but a dash is longer and it is used differently.

So you can use it, this site explains how:

In informal writing, em dashes may replace commas, semicolons, colons, and parentheses to indicate added emphasis, an interruption, or an abrupt change of thought. [...]

Solution 2:

I think that would qualify as a "dash" rather than a "hyphen". It seems to have a legitimate role in punctuation, to signify the breaking off from one train of thought, to start on another.

Of course, it's easy to overuse such a device. You don't want people thinking your thought processes are always jumping around all over the place!

Solution 3:

The symbol you used in your example is indeed a hyphen, but you are using it to function as an em dash. So technically, you're using the wrong symbol.

The hyphen ( - ) should only appear in compound words and phrases (as in, eight-year-old, ex-wife, stick-in-the-mud, a part-time job, etc.).

The em dash ( — ) is best explained by the Chicago Manual of Style: Em Dashes are used to set off an amplifying or explanatory element and in that sense can function as an alternative to parentheses (second and third examples), commas (fourth and fifth examples), or a colon (first example)—especially when an abrupt break in thought is called for.

• It was a revival of the most potent image in modern democracy—the revolutionary idea.
• The influence of three impressionists—Monet, Sisley, and Degas—is obvious in her work.
• The chancellor—he had been awake half the night—came down in an angry mood.
• She outlined the strategy—a strategy that would, she hoped, secure the peace.
• My friends—that is, my former friends—ganged up on me.