L’autre semaine refers here to the week before the current one, or, depending of the context, the last week these people talked together.

L’autre is versus the current one.

The expression “l’autre jour” means “some day in the recent past”. Despite the definite article, it does not always refer to a specific day. It’s an idiom. When the day hasn’t been mentioned before, it’s more or less interchangeable with “il y a quelques jours” (“a few days ago”). However “l’autre jour” also works when some additional qualification is given. English has a similar idiom: “the other day”.

J’ai croisé Marie l’autre jour.   I bumped into Marie the other day.
J’ai croisé Marie il y a quelques jours.   I bumped into Marie a few days ago.

Quand je suis passé à Paris l’autre jour, il faisait beau.   When I was in Paris the other day, the weather was nice.

This construction can also be used with longer time periods, but it’s a lot rarer. So “l’autre semaine” (rare) means a few weeks ago, “l’autre mois” (very rare) means a few months ago, etc. This diagram compares the approximate relative frequency of “l’autre jour” among the uses of “jour”, with the same relative frequency for semaine and mois.

This construction can also be used with recurring events: “l’autre fois” = “the other time” = a few occurrences of some event ago. Sometimes “l’autre fois” refers to the last occurrence before the present (i.e. it’s a synonym of “la dernière fois”), sometimes to an earlier (but not much earlier) occurrence. Similarly, “l’autre soir” means a few days ago in the evening, “l’autre matin” means a few days ago in the morning, etc.

If you use the indefinite article, then the meaning changes to some future occurrence that is not yet determined.

Nous en parlerons une autre fois.   We’ll talk about this some other time.
Je reviendrai un autre jour.   I’ll come back some other day.

Thus, in the sentence from Patachou, it is implied that the two people having the conversation talked a few weeks ago. During this earlier conversation, the speaker told the other person about a little boy.