"We went swimming later in the afternoon, Jack and I."

Solution 1:

We went swimming later in the afternoon, Jack and I

This sentence has been done something to.
It's an example of the syntactic rule of Right-Dislocation.
The sentence it's transformed from is

• Jack and I went swimming later in the afternoon.

The rule copies an emphasized Noun Phrase (which may be subject, object, or oblique) in a sentence, and repeats it, with a different intonation, for emphasis, at the end of the sentence. It's not a movement rule, but a copying rule, since the original NP remains in place as a pronoun.

There's also a rule of Left-Dislocation, which copies the NP to the beginning of the sentence.

Here's the entry from Haj Ross's list of The Top 200+ English Transformations
  (p.4, categorized under "I. Emphasis; A. Pseudoclefts and Dislocations")

"6. LEFT AND RIGHT DISLOCATION:

• My horse snores. ➞ My horse, he snores. (via LEFT DISLOCATION), or

• My horse snores. ➞ He snores, my horse. (via RIGHT DISLOCATION)

In pseudoclefts, this rule will produce related sentences like the following:
Anne's brother left ➞ Anne's brother is the one who left ➞ Anne's brother, he's the one who left."

Some more examples of dislocated sentences:

• My Uncle Will hates the Dodgers a lot. (Base sentence)
• My Uncle Will, he hates the Dodgers a lot. (Left-Dislocation of Subject NP)
• The Dodgers, my Uncle Will hates them a lot. (" of Object NP)
• He hates the Dodgers a lot, my Uncle Will. (Right-Dislocation of Subject NP)
• My Uncle Will hates them a lot, the Dodgers. (" of Object NP)

As for why anyone would think any of these are more or less "formal" or "old-fashioned" than others, I can't really say. "Formal" and "old-fashioned" are not linguistic terms, anymore than "fad" or "fancy". Everybody has their own idea(s) about these terms.

Solution 2:

• "We went swimming later in the afternoon, Jack and I."

That's a right dislocation construction. It is common in informal style (and usually the pronouns in the dislocated noun phrase will use accusative case).

CGEL page 1408:

• Dislocation of this kind is often found in oral personal narratives and informal writing.

There are good pragmatic reasons why speakers and writers use dislocation. In some contexts, the dislocated versions have advantages over the more basic non-dislocated versions.

The reason why your example sentence seems a bit strange or awkward sounding is probably because the personal pronoun in the dislocated phrase is in nominative case, which is rather unusual (though, if this is fiction, then the author could have done that intentionally to stay consistent with the narrative voice).

Here's a related tidbit. CGEL page 462:

Accusatives are also the only option for the left- and right-dislocaton constructions (which are themselves characteristic of informal style):

[18]

• i. Me, I wouldn't trust him further than I could throw him.

• ii. I don't much care for it, me.

If you're interested in more info about these kinds of information packaging constructions, you might be able to find it under the topics of right-dislocation (and left-dislocation).

Note that CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL).