I would also ask similar questions — see if you can spot the trap —
when to tenuki ? when to invade ? when to reduce ?
when to cut ? when to jump ? when to extend (nobi) ?
when to connect ? when to push ? when to hane ? when to connect solidly ?
when to fix with a tiger's mouth ? when to not fix at all ?!
when to attach (contact enemy stones) ? when not to attach ? ...
( the list is almost endless. )

This is the trap: I was looking for some general, nebulous
one-size-fit all answer. Because that would be so convenient !
Learn it once, apply forever, in many ( or ALL ) situations.

There is no such thing in Go.

A generic one-size-fit-all reply: when it's good for you.
Another version: it depends. ( Depends on what, you may ask. Reply: Exactly. )

Other common variations: when the trade is good for you;
when you gain more territory than your opponent;
when you gain more power than your opponent;
when good aji is more important (than other factors);
when it's better for yose (later);
when you cannot lose the cutting stones;
when you should just give up the useless stones;
when eyespace is more important (than other factors);
when no ko threats is more important (than other factors);
... the list goes on endlessly, again.

Adults ask questions like these. Adults love general patterns, general guidelines. Proverbs.

Kids don't (usually) ask these questions.
A teacher would show a double approach to a child; the child accepts it.
The next time, the teacher shows another double approach in another situation; the child accepts it.
Yet another time, the teacher shows when a double approach is bad in another situation; the child accepts it.
Over time, over tens and hundreds and thousands of different situations, the child gains experience.

This is how a child learns to walk. The child does not ask, "When do I put my left foot forward?"
"When do I put my right foot forward?" "When do I skip (instead of a regular step)?"
"When do I stop (all together)?!" "When do I step sideways? "
"When do I step backwards?" The child walks, and falls -- rinse and repeat.

This is a perennial discussion -- many many threads, thousands of words (and hours) have been spent
on this and similar topics here on this forum, on KGS, etc.
Some people would raise the point that learning to walk or to talk is fundamentally different than learning Go.
Others would point out that some general guidelines are useful for beginners;
or, when we reach a new "beginning phase," which happens repeatedly in our growth curve.

That's all good.

But the bottom line is this: in Go, the exceptions are the rule.

Corollary: the first (and only) rule in Go -- There is no rule. (Same in the martial arts. )

I'm sure Bill (and others) will have their brilliant replies, other ideas. This is my take, from my own experience,
and from watching the teachings and experiences of many other people.

...and good point: (in addition to the same post 15 about traps,)
playing overly defensively with your handi stones is not a good way to use them (not efficient).
An efficient way to use your handi stones is to attack.
This is a generic, one-size-fit-all reply -- to figure out when to attack, when to defend,
with or without your handi stones, you need tons of experience, actual Go combat experience, and reviews.

Supplement to Post 15:

There is at least one pro, a few years ago
she did a series of online audio lectures on the double approach.

Something like 12 to 14 lectures, each about 40 to 45 minutes.

I estimated, as an adult, with a job and family, etc.
a reasonable amount of time would be about one month per lecture.
To reach a level — let's call it, "not entirely clueless."

So, to get a very rudimentary idea about
the double approach from this series: about one year.

This is another angle to your original question, " When to double approach? "

When we start, whether as an adult or a child, we lack experience.
This is one reason adults look for guidelines, general patterns, and proverbs:
because from our own experience in life and in other areas,
we've figured out lots of general guidelines and "wisdoms" that are very helpful for ourselves.

That's OK. But a better understanding also requires actual experience.
Tons of it. Because we need to learn when the rules apply,
and when they fail miserably. This takes time, experience,
and hard work (study and review). Eventually, we learn to
un-learn the rules.

A short question: "When to double approach?" —
this appears to be a simple and reasonable question,
why is it so involved (difficult) to answer ?

Short reply: because of the immense number of variations.

Longer reply: Before we can even begin to look at
the double approach variations, we think about the pre-requisites:
what kind of background knowledge do we need ?

• Basic contact fighting skills. This is endless, but:
  (*) one year's worth of study and experience is a good start.
  
• Basic shapes. Same note as (*).
• Basic tesujis. Same note as (*).
• Basic life-and-death. Same note (*).
• Basic assessment of the board. Same (*).

So, one can try to answer your questions about the double approach,
(and others will), but it's not so easy to include
the amount of material in this thread.

( Example: the number of likely life-and-death variations
arising from the double approach is in the
hundreds or thousands — I'm not surprised if millions or more. )

All the same, no better time to dive into
studying the double approach than right now! — that's my opinion.