If you're not comfortable with having an app comb through your inbox, you might prefer an app you manually control instead. Parcel (free with in-app purchases) is another tracking tool, but this one just focuses on shipments and deliveries--you won't find any budgeting tips here. Just enter your iPhone's shipping number as soon as you get it from Apple, and Parcel will hunt it down. (You can also scan a package's barcode.) Though UPS is Apple's standard carrier, Parcel supports shipping info from more than 240 shipping companies worldwide.
Parcel will display your shipment's up-to-date info within the app, and also in the Today tab in Notification Center if your device is running iOS 8. Tap on an item to see where your shipment is in the app's map view. However, if you'd like to receive push notifications, that will require you to cough up $2 to get the premium version of Parcel, which also lets you track more than three shipments at a time.
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