how to avoid the plus-que-parfait

Question

I want to translate the following into French:
When I had signed the letter, Maurice mailed it.
I would write this:  "Quand j'avais signé la lettre, Maurice l'a envoyée."
What I want to do is avoid the pluperfect in the subordinate clause. l reached for the usual stand-by "Après avoir/être + past participe" but that will not work here, as there are two subjects (I have always used "Après avoir ..." in the context of there otherwise being one subject) and "Après avoir signé la lettre, Maurice l'a envoyée" means "After having signed the letter, Maurice mailed it" but it also means that Maurice both signed it and mailed it.
 That is inescapable because "Après avoir/être ...", a hanging participle/dangling modifier, lacks a subject, which is provided in a sentence by the first noun in the main clause, thus there is no doubt as to who signed it - it's Maurice because of the operation of that principle. (At least, that's the case in English; if French is somehow different, please tell me.) 
I then tried "Après que ..." but I still end up with the pluperfect because inescapably the signing of the letter is complete and precedes the posting of it by Maurice.  Then I thought to use the passive, but that seems generally to be anathema to the French and it is really clumsy to try it here.  Then I thought to change the exact English formulation of the sentence and use "J'ai signé la lettre et puis Maurice l'a envoyée" but I don't think that that helps (nor is it correct). The signing precedes and is complete before Maurice's action and I keep ending up with the obvious - passé composé for Maurice's action and plus-que-parfait for the action that preceded it (signing).
 
On reflection, I suspect that I can only resort to the passive to solve this problem:
Maurice a envoyé la lettre qui avait été signé par moi.
(It doesn't matter to me that the passive is anathema to the French; this is just part of a grammar exercise.)
Do you think that that is correct or am I missing something in my quest?
Any solution please, other than the passive, to re-write the original in a way that avoids using the pluperfect in the subordinate clause?

Accepted Answer

Bonjour Le P.
This is an interesting question (although I'm not sure it's part of a Kwiziq exercise... ;-) )!
In this instance, to avoid Le Plus-que-Parfait, you should use Le Passé Surcomposé as it has similar value to Le Plus-que-Parfait and it is far more elegant. Also, using "une fois que" instead of "quand" is far better too.
Une fois que j'ai eu signé la lettre, Maurice l'a envoyée.
I hope this is helpful.
Bonne journée !

Answer

Just to add to this very interesting discussion:
The passé surcomposé is a compound tense in French that is less common than other past tenses but has uses in certain situations. 
It is generally considered to belong to a more formal or literary language register.
Its use is more common in certain French-speaking regions, notably in Switzerland.

Answer

Thank you, Céline and Alan G.  My question arose from some practice exercises in a publication that either did not include answers or that I failed to extract at the time.  Although it did not arise directly from something on Kwiziq, the problem that it posed is relevant to French grammar on Kwiziq.  My quest to work out the answer (which failed, but for your contributions) led me down a path that reinforced my understanding of the inevitability of pluperfect+perfect, which I have found most useful.  Your answers have quietened my mind and my frustration at not being able to solve the problem to my satisfaction.  I can now rest easy, as they say, because I have answers from reliable, trustworthy sources instead of the risk of half-baked and inaccurate information from the internet (including some dictionaries/translation applications there. Thank  you both.