I have heard that the "me shebayrach" for cholim or government, etc. should be said before the maftir, and not between the maftir and haftorah, why?
ב"ה Shalom I Haven't found any particular reason, if your Bet Knesset has such a particular custom. There is a Halacha not to speak during Kriyat hatora, even between aliyot and even Divrei Torah. ((See שולחן ערוך או"ח סי' קמ"ו). Making a "me shebayrach" is not considered speaking and it is the worldwide custom to say "me shebayrach" then. In the Sefer אז נדברו, ח"ט סי ז,, Rav Zilber zt"l was asked if between Kriat Hatorah and Maftir has the same law in regard to speaking as the other Aliyot, and he answered that it seems to him to be the same . Rav Efraim Greenblat zt"l in ח"ג סי' קצז, רבבות אפרים relates to this answer of Rav Zilber zt"l, and says that no consideration was made in regard to between maftir and haftorah. In any case, in regard to speaking this seems to be no difference. I would say, that what you heard, may be just because there are a few "me shebayrachs" One for the maftir , and another for the Magbihah and then another for the Golel, and if more are said it would bring to possible talking in Shul. All the best