No return offer due to "headcount", how to rerecruit and frame my story?

As title reads, I was just told that I wont get a return offer from my off-cycle internship. 

 I asked the person responsible for recruitment at my bank what to say when re-recruiting, and was told that I could use "No headcount" as a reason for not returning/signing with the bank, and that the bank would back it up. However, he also said that there were some improvements areas so even if there would have been clear headcount, it wasn't certain that I would get an offer, and that it were unlikely I would get hired in the future if headcount opened up. 

Further, I spoke to a Director and an MD, whom both told me, without me asking or even hinting at it, that they would both happily give me a good reference if ever needed. The Director even said that if someone calls him, he would say that he thinks I should have been hired and said that we should stay in touch going forward. I also talked to the associate I worked the most with, who told me that he would happily give me a good reference and that he don't have anything "bad" to say about me. I truly believe they are genuine, as I have a very good and open relationship with all three.

Next up I am going to do a fairly long internship at a buy-side shop, but they do not hand out return offers to undergrads so I will need to re-recruit with other banks in the future. 

Now I am wondering how to best frame my "story" in future interviews? Feel its a bit hard to clearly get my head around as I obviously didnt get an offer, but will be able to "blame" it on headcount, while knowing the seniors are very genuine when they say that they will give me good references. 

Given that they will give good references, would full-time recruiting be possible or am I better off aiming for another internship? And how much do I wanna push references etc when interviewing?