This is possibly the WORST advise an Army recruiter can give. I try not to step on the Recruiting Command's toes, but for years and years they have been submitting packages in June, July August and September. Many schools start in the June/July timeframe. It can take anywhere from three weeks to three months for the scroll to come back from Sec Def (depending on his schedule). Add in the possibility of needing a medical or moral waiver, missing documents, documents not signed, not dated correctly and you have the recipe for a very frustrating first military experience.
PLUS we are now dealing with the Direct Commission Course. We can send you before graduate school starts, allowing for BOLC-B between M1 and M2 which is the optimal time. BOLC-B any later while in HPSP is, problematic. BOLC-B after HPSP means reporting to PGY-1 in August instead of June.
Remember, you can't get any scholarship benefits until your packet has been forwarded to the HPSP Student Management Office. It is your best interest to get your recruiter to get your packet in and approved not only so that you get your stipend and tuition paid on time but so that you can fulfill the necessary military training in the optimal timeframe.