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I just came across a very nice example of a rhetorical structure I know I have seen many, many times:

Our national character feels like it’s possessed by every hellish ghost of American history: white supremacist patriarchs, gilded age swindlers, paranoid McCarthyists, Know-Nothings and Klansmen and con artists and terrorists.

(Source: Sarah Kendzior writing for The Globe and Mail.)

Notice that the usual comma-delimited list turns, after the word "McCarthyists", into an unpunctuated list separated by repeated use of the word "and". The effect (at least on me as a reader) is to create a simultaneous sense of both urgency (we are trying to get through this list as fast as possible) and exhaustion (this list is SO LONG that we can't even use commas any more).

Does this rhetorical technique have a name?

You are likely thinking of polysyndeton and asyndeton—probably the latter, but the former can achieve the same result in a different way. From your example, the list of which includes both comma-separated items and conjunction-separated items, I might conclude both could be applicable here.

Polysyndeton is the use of multiple conjunctions to stretch out a passage of prose to avoid coming to a full stop:

I said, "Who killed him?" and he said, "I don't know who killed him but he's dead all right," and it was dark and there was water standing in the street and no lights and windows broke and boats all up in the town and trees blown down and everything all blown and I got a skiff and went out and found my boat where I had her inside Mango Key and she was all right only she was full of water.
— Ernest Hemingway, "After the Storm."

Asyndeton is the avoidance of conjunctions, "often resulting in a hurried rhythm or vehement effect":

I have found the warm caves in the woods, filled them with skillets, carvings, shelves, closets, silks, innumerable goods. — Her Kind by Anne Sexton

(See Sylva Rhetoricae.)

Ward Farnsworth, in Farnsworth's Classical English Rhetoric, refers to this simply as "asyndeton […] with polysyndeton". (And the only reason it's asyndeton with polysyndeton, rather than polysyndeton with asyndeton, is that polysyndeton is the chapter before asyndeton, so their combination is discussed under the latter.) So I don't think there's a name for this.

That said, of his three quotations showing asyndeton followed by polysyndeton (page 154) and one showing polysyndeton followed by asyndeton (page 155), none is quite like your example, where a single list changes partway through from one to the other. Of his four, the one that comes closest is this:

From a national and imperial point of view, you need never be alarmed at the dangers of one-man power so long as the House of Lords endures. Be he minister, be he capitalist, be he demagog – be he Mr. Gladstone, or Mr. Chamberlain, or even Mr. Schnadhorst – against that bulwark of popular liberty and civil order he will dash himself in vain.  —Spencer, speech at Birmingham (1884)