Tizzy
If you were playing an OCC scenario game the rules may have been changed to allow you to disband.
On standard rules, if you have one, size one, city and it produces settlers it does not disband and you do not lose any food from the food box.
There are two situations where you can use this to get a fast start. One is when you get 100 gold (sometimes 50 is enough) from an early hut. You just switch to settlers and rush buy before getting to size two.
The second is more satisfying. If you get a river start, build on a rivered plains/grassland and work a forest square - preferably rivered.
Your city will accumulate food much more slowly than it will accumulate shields in the production box. Set production to settlers (warrior first if you are timid) and build those settlers in the usual way. The result is that the settlers are produced and your second city gets established earlier than it would otherwise have done. The loss of food production to help your capital grow is ameliorated by the fact that no wheatsheafs are lost when the settlers are completed as they are when the capital has already reached size two before the settlers are completed.
You're really cooking with gas if, while the slow growth capital is working on the first settlers, you acquire some gold. Now you wait until the first settlers are built and until the capital is about to get to size two (moving the settlers into position ready to found and switching to work a plains/grassland square so as to feed them in the meantime) then use the money to rush a second settlers. You still lose no food and don't disband. Next move the second set of settlers appears, the capital goes to size two and you promptly found your second (and shortly afterwards, your third) city.
If you have a silk special to work (or, ideally a rivered silk) then you will not only get off to a flying start territorially but your early research will rock too! Rivered pheasant is not too dusty either.
One (minor) problem. I have got an advanced tribe outcome from a hut while following this tactic. Now you are going to have to switch away from settlers and may give up a bunch of shields doing so (once you have a second city the capital reverts to disbanding if it builds settlers while size one). In this case the advanced tribe is unlikely to be as well located as the site you had in mind for your second city. But the advanced tribe has still given you a big early boost so this is not really a drawback.
Another small plus is that while you stay at size one and have no other city established, any unit will successfully defend the city against any number of barb attacks. Twice this led me to capture a barb chief with just a warrior in my capital.