1) Start with any performance you want to add the Mark V flange PART,
2) Press [+] from the home screen to add a new PART, select "Search" to search for what you want vs. typing in a number.
3) At this point, there is information overload - so narrow down your search by choosing the "Bank" which is your Chick Corea library. When you choose the drop-down menu from "Bank" there will be a list and your libraries will be included in the list. It's possible you do not have the Chick Corea material in a library (as you've imported this to User). If this is the case, then you would pick "User".
4) If you picked the library, then you will be presented with a list of all PARTs that made up the Chick Corea set. Make sure the category is "All" or the category will narrow down your search. You can scroll around to find what you're after - should not be a ton of PARTs to sort through.

I'm saying to do it this way because it's possible your memory of what the PART is called is incorrect. In which case a search would not find it.

Or, you could:

1) same
2) same
3) Click on the magnifying glass (upper right search box). A text entry menu should pop up - type in "Flange"
... every PART matching the word "flange" will show. Even with "all" selected for Bank - there should not be a lot to look at. The presets have 5 "flange" matches.

If you do not find it, then what you're looking for doesn't have the word "flange".

When you recall the performance that's the flange performance (not part, performance) - look at the [PERFORMANCE] (HOME) screen and take a look at what the PART names are. Namely the 1st PART. If it follows the preset convention - it should be the same as the performance name but doesn't have to be.

Are you sure it's "Flange" you're looking for and not Phase?

The new set (I'm not sure if you're using the Motif XF or Montage set) has a performance:

"Chick`s Mark V phase"

So for that one, you'd look for "phase" not "flange".

1 fried brain and 2 days later I stumbled on the performance that had this part I wanted and realized that while the performances are well catalogued there are no breadcrumbs to connect the pieces (parts) that make up the performances. Is this true or did I miss that in the manual somewhere?

Jared,
It is not stated in so many words in the Manual.
But, as I'm sure you know, a Performance can contain as few as one and as many as sixteen Parts plus an A/D Input and Digital In Part. That's the definition of a Performance.
All but one of the 2036 Factory Performances contain 1 to 8 synth Parts.

Those that are Single Part Performances as you know can be "merged" and Multi Part Performances can be "merged" into playable eight Part Performances.

If you begin to think of a Performance as the instrument's sounds you need for performing, it can help. You can combine several Performances to make a new Performance but you also can go and get any single Part from any other Performance for use in your new Performance.

Say you like the FM EP from "CFX + FM EP" - you probably know by now that this is the FM-X electric piano in Part 5 of the first Performance... when constructing a User Performance of your own, you can go can get just Part 5 from that Performance.

When exploring a Factory or Library Performance, you should look at each Part, see if it is a standalone component or is it a construction component... there are Parts that are just organ leakage, percussion, hammer noise, pitch scoops, and then there are Parts that are complete playable instruments. As you'll appreciate as you begin programming your own, it's programmer's choice about the role each Part plays...

I think your current issue is you forgot the name of the Part. Had you remembered, your search would have been successful.
You can (and it is highly recommended that you) name each Part that you create... this makes them easier to find when looking to reuse it in another Performance.

Silly me...it's phase. Ok, forget that...here's another question; I copied Chick's Mark V library to user but the LIve set for that library didn't make it into any of my Live set banks. What'd I do wrong?

You may not have done anything wrong. Although there is a feature to copy live sets between banks - I'm not sure how this could possibly work for Library to User. I haven't used the feature - but there may be limitations to what you can do in terms of Library Live Set saving (copy/exchange).

Someone with more experience with this new feature should chime in.

If you cannot preserve a live set (I'm thinking you cannot) - they are not difficult to construct. All the data is automatically filled in when doing a manual add and is reasonable without editing.

Jared wrote:

Silly me...it's phase. Ok, forget that...here's another question; I copied Chick's Mark V library to user but the LIve set for that library didn't make it into any of my Live set banks. What'd I do wrong?

Nothing... if by "copied" you mean you Imported the Library to the User Bank - everything but the Live Sets will be Imported. The reason becomes clear as IMPORT LIBRARY is used generally to create a new configuration... either edit and update an existing Library, or combine selected Performances from various Libraries... with the end result being a new Library. Since LIVE SETs point to the exact location of the data, the act of Importing means you're going to be changing the location.

From MUSIC PRODUCTION GUIDE 04/2017: "Montage Library Import - Combining Complete Libraries"
"If all Performances are selected in the Library Import display with "Select All", a complete Library including all Performances, User Waveforms, and User Arpeggios is copied into the User Bank. Only the Live Sets contained in the Library are not taken into account."