I could, after a fashion. Your stock cell phone is not designed to be a efficient repeater. If you need the capability, buy a device designed for that.
Ignoring for a moment the red flags this question raises, given that actually making such a device would be entirely in contravention of FCC (and ever other regulatory agency on the planet) regulations, it is not to hard to figure out what this is not feasible.
The phone has to be able to send and receive signals simultaneously. It receives the cell site signal and transmits its own. These are two different frequencies and the hardware of the phone receives what the cell sends and sends to it.
This means the receiver and transmitter are backwards from what a repeater would need to do—receive on the phones frequency and transmit on the tower's—for the phone—and in the case of the link to the cell site, it would need to behave like a normal phone.
This entire approach is much too complicated but it is the only way you could create a repeater with a phone—more likely two phones. Even then... it would be a foolish approach.
In practice, cell phone booster a simply bidirectional RF amplifiers. This is much simpler and while you could scavenge some parts of a mobile handset to make one, you would need many other parts and it would be easier and cheaper to just buy what you would take from the phone. Even in a Mad Max world, trying to reuse components from a cell phone for anything but repairing another cell phone, makes little sense.