Here is an alternative approach:

The first thing that I would like to point out, that I think is very important and perhaps was missed in this whole conversation, is that shelter stays do not add Housing History records. There are, in fact, two different types of records. There are Stay records (added via the Admissions module) and there are Housing History records (added via the Housing History module).

A long long time ago, shelter stays were only displayed on the Admissions screen and housing history records were only displayed in the Housing History. I advocated for Admissions records to create a corresponding Housing History record, and was told by the developers why that was a Bad Idea. They asked what the goal was, and I stated that the goal was for users to be able to see the Housing History records and the Admissions records all in the same place. They pointed out that a better solution would be to simply have Admissions records listed on the Housing History screen. This was a far more efficient and less error-prone approach to accomplish the desired goal. And this was ultimately the solution that got implemented.

Fast forward several years, and now housing status is automatically calculated from a combination of the client's HH and Admissions records. Now, I think the better solution is for Encampments to also inform housing status, much like Admissions does, without creating a housing history record.

Regarding the priority sequence for Housing Status, I'd suggest that the Encampment Enumeration takes the highest priority - if someone was found to be in a tent today, they're definitely not anywhere else. I know that some shelters have in the past given clients like a "weekend pass" or something where they get to keep their bed but don't have to sleep in it (I hardly agree with that approach, but it is done), so they don't get booked out but they're not in their bed. And in some places Admissions are used for motel stays, and it may not be known whether a client is actually present at the motel room that is being paid for or not. So then Admissions are the second priority, higher than Housing History, which is the fallback if nothing else is present.

Some discussion here:

One way to do this is via Housing History. An Encampment enumeration record creates automatically a Housing History record for a client. However, this can be problematic because the fields in an Encampment Enumeration don't match the fields for Housing History.

An enumeration is instantaneous while a Housing History has a duration, so what is to be done about the start and end dates? If the client is enumerated on Tuesday and then again on Wednesday, should that create two separate 1-day Housing History records, or one 2-day Housing History record? What if the client was enumerated on Tuesday, there was no enumeration done on Wednesday, and then the client was enumerated on Thursday?

What if the community has made the Rent field on Housing History mandatory, or the Address field, or the Landlord field?

It would be problematic and inaccurate to create a HH record based on an EE record, not to mention inefficient - the data is already there, so why add a second copy of the same data?