From Li-Ion to Li-Po battery: design a new battery pack for old mobile phone
The most important aspect is the charging voltage. You must never charge a Lithium battery (Ion or polymer) above the voltage at which it is rated to charge. So the charging voltage must be the same or below for the new package. 

The big question is what is the charging voltage of the charger in the phone. What you stated is the battery charger output of 7.5V @ 880mAh is unlikely to be the charger itself. There is likely a charger IC inside the mobile phone. Perhaps you can measure this by measuring the voltage of the pins of the battery with the battery inserted. For 1 cell this should be 4.1V, 4.2V, or 4.35V more than likely.

As for current, usually don't charge battery above 1C (where C is nominal capacity). So make sure the battery capacity is larger than the old battery. Check the datasheet of whatever battery you buy about this.

Unless you use the original battery, your solution is unlikely to prove optimum, as far as extracting the maximum amount of charge, but this shouldn't be an issue.

[Edit]

As I mentioned above, looking now at your picture, it seems U2 is the charger IC. It'll be good for you to obtain its markings and perhaps the datasheet.