Once you have an online bank such as ING Direct, Emigrant, or any bank for that matter; they will allow you to set up different banks that you can transfer money from. The key might be in you getting the savings account first at another bank. I had a similiar problem with my own bank and I realized the beauty of having an online bank somewhere else.
As for the savings, you can deposit as little as a dollar in some. Don't be too concerned about the small amounts; they grow quickly.
Also, HSBC will give you an ATM card that you can use locally to add or withdraw money from.
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