Is A Two-Week Notice the Only Way to Not Burn Bridges?

Basically, I just got a job and they would like for me to start next week. I don't have any beef with where I work now, nor do I have any desire to leave on poor terms. My thought was to still do the letter and give the manager(s) the verbal head's up, but most of the articles I've been reading is that giving any kind of short notice will burn a bridge and potentially mark one as "non-rehirable." I'd prefer to keep this organization (large corp where I am still entry-level) as a place I could return to if crap hits the fan, but don't know what I should do. I worry that asking the new place (a small business) to change my start time would look bad/like a red flag.

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TIA for any feedback!

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ETA - In the USA in a "right-to-work" state.

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