My impression is that, indeed, the second sentence is a little bit odd; but very slightly so, so that in everyday language you would probably not pay attention. However, in a more formal context where correct use of language is crucial (e.g. when writing a novel), you would seek correct use of tenses, e.g.
Nachdem ich Besuch von meiner Familie gehabt hatte, ging es besser.
Nachdem meine Familie zu Besuch gewesen war, ging es besser.
You may find it interesting that nachdem is sometimes (and in some regions) also used as a synonym of weil or da, as in
*Nachdem ich jetzt schon mal hier bin, kann ich auch noch mit dir Kaffee trinken. [ODD]
This is not correct by the rules of standard German, and you should not use it; but you may encouter it in everyday situations.
You could interpret your second sample sentence that way, too, at least theoretically;
*Nachdem [= weil] ich Besuch von meiner Familie hatte, ging es besser. [ODD]
which then would imply simultaneity of the two events ("Family visiting" and "things go better"), not sequency: things went better at the very time when the family was there.
Yet, this is most probably not what the author had in mind because hardly anyone would use nachdem in the sense of weil this in written language. It is just technically possible to (mis)read that sentence that way e.g. when you assume it is part of an oral conversation.
With the correct tense gehabt hatte, such misreading is unlikely:
*Weil ich Besuch von meiner Familie gehabt hatte, ging es besser. [ODD]
In order to make this sentence fit a real-world situation correctly you would have go to the lenghts of constructing a really complicated cluster of time periods in the past part of which ended in the past and part of which continue to the present.
For example: a person reports about a continuous situation in the past (Es ging mir besser), and relates this by the way of cause and effect to a previous event which however finished prior to the onset of the continuous situation (Weil mich meine Familie besucht hatte).