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From what you are describing, you have the following network topology:
Head office - Private Network - Branch offices
It seems what you want to do is have local PSTN access at all offices and use your own private network as the backbone to transport calls between offices and to the PSTN for toll bypass. All that is required here is to have IP to PSTN gateways at each office. The type of connection to the PSTN at each office will depend on the call volume. For small offices, a number of analog circuits may be sufficient. For medium size offices, a fractional T1, T1, or multiple T1 circuits may be sufficient. A T3 is equivalent to 28 T1 circuits, so unless you're moving a very, very large number of calls, that won't be necessary.
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