This is the first article in a small series I hope to write on the subject of הושענות. The series aims to cover the halachic, minhagic, and liturgical aspects of הושענות. I do not plan to offer in-depth explanations of the meaning of the הושענא prayers themselves. For that there are many wonderful works available elsewhere. Just a disclaimer.
We are נוהג to bring a ספר תורה onto the בימה and encircle it once each day, and on the seventh day we encircle it seven times [1] זכר למקדש. In the בית המקדש the כהנים would conduct a procession around the מזבח once each day while saying אנא ה הושיעה נא אנא ה הצליחה נא [2]. (According to רבי יהודה, they would say אני והו הושיעה נא [3].) Today where, בעוה"ר, there is no מזבח, so the חזן (or other individual appointed by the ציבור) stands at the בימה holding a ס"ת, and the congregation encircles him, similar to what was done by the מזבח. The above was done every day of סוכות. (On הושענא רבה, however, the כהנים would encircle the מזבח seven times. To that end, we encircle the בימה seven times on הושענא רבה as well. This will IYH be covered as a subject of its own in a future article.)
Being that, in the בית המקדש, the מזבח was encircled following the קורבן מוסף (as evinced by the continuation of the משנה in which the כהנים would take leave of the מזבח - indicating that there were no more קורבנות to be brought), therefore we also encircle the בימה following מוסף [4].
רב סעדיה, however, thought that we should encircle the בימה immediately following the הפטורה while the ס"ת is still at the בימה, so as not to take the ס"ת out of the ארון unnecessarily. Perhaps רב סעדיה holds that הושענות have to be done as early as possible, even before מוסף (similar to נטילת לולב which also has to be done as early as possible, even before מוסף). To that end, רב סעדיה would understand the משנה to mean that the כהנים would take leave of the מזבח as the formal end of the ceremony, but not that they were finished for the day necessarily. The טור, however, reports that we are not נוהג that way. The reason for this, as the בח explains, is that our entire מנהג is only זכר למקדש, therefore we are actually acting properly where we delay הושענות to the end of the service, as in the plain understanding of the משנה [4].
שער הכוונות, however, says that the right place to do הושענות is after הלל, which was always the מנהג of ארץ ישראל [5]. While שער הכוונות does not offer any halachic reason to do so, nonetheless אגרות משה [6] writes that if one already is holding the ד מינים, then to put them away and then take them out again later looks like passing off a מצוה.
[1] שו"ע או"ח תרס:א
[2] משנה סוכה ד:ה
[3] This is a request of the One Who is Himself (הוא) with me (אני) בצרה to offer salvation from all the troubles that surround us in the merit of the תורה.
(ע"ד עמו אנכי בצרה; ואפשר שזה ג"כ כוונת דברי התפא"י שם באות כה)
[4] בח או"ח שם
[5] כף החיים שם אות ד. According to this, קדיש תתקבל is then said after הושענות, rather than after הלל.
[6] חלק ג סימן צט. This is in addition to זריזין מקדימין למצוה (מקראי קודש סוכות סימן צ). See also בכורי יעקב (תרס:א) who proves that, actually, even in the בית המקדש they would say הושענות before מוסף, and where the משנה references taking leave of the מזבח, the משנה does not mean taking leave of the מזבח after הושענות per se, but rather taking leave of the מזבח in general, after all the עבודות of that morning were finished.