How much do you need? I think there's a strap of $2`s in the safe (older, but US currency does not expire) and I usually keep others around for emergencies.
At the same time, I'm struggling to come up with a scenario where it is possible to enter the US legally and not encounter a business or person that can make change. From your wording, I guess it's possible that you could be visiting a place outside the US that will transact in US dollars and is perhaps economically depressed -which could explain the low denominations. If so, the 2$ bills may be frowned upon or taken as a joke.
SBB offers to deliver cash according to your travel needs. Of course, there's a commission and they offer delivery times between 24-48 hours. Not exactly cheap, but it works.
I only used them several years ago for MXN and they delivered. Then I learned I can visit the ATM at home and keep a bit of emergency cash for next travel.
You've probably been to the bank by now, but I'd be curious to know the answer you received. Last time I asked for USD at MigrosBank, it came in 20s and larger. It didn't occur to me to ask for smaller bills, so I don't know whether they can do it.
Because people may have experience of those places and be able to give real help*
I've been to countries where I have been told it's impossible to change money easily, or legally, and then by that evening I've got hold of a nice wad of US one dollar bills at a favourable exchange rate.
*Also, we're still intrigued by the high altitude / sea-sickness thing!
It really wasn't a straight forward question that you asked. You asked roughly the same question multiple times in different ways and added extraneous information which only served to distract the reader.
As an adult, you could have called Migros bank yourself and asked them directly. If answer not adequate you could have come here and simply said:
"Where can I get smaller denominations (1,5,10) of US dollars? I called Migros bank and they do not offer this service."
As an enhancement, you could have added that you needed 500, 1000 or 5000USD or whatever.
I'm guessing that there was no fee, because there was no currency exchange - it was simply an exchange of large US dollar bills for smaller US dollar bills. I was going to suggest a currency exchange either at the Bahnhof or at the airport.
It's win-win. The currency exchanges don't like the hassle of having lots of small bills on-hand (especially in Switzerland, where people tend to exchange relatively large sums of money). And I understand what Jamboree is trying to achieve - there are plenty of countries that use the US dollar as parallel currency, but dollars are in short supply in those countries and you can't count on someone being able to make change. So you travel with lots of small bills so that you're able to pay with exact change.