Without the tracking number, you won't be able to receive the order information; however, you may use Informed Delivery
by USPS. This service allows you to preview parcels before arrival via gray-scale photographs of the items and their labels. It also allows you to manage packages and track all the shipments without the need to enter individual tracking numbers. To take advantage of this service, you will need to create a personal account, verify your identity and address, and choose the devices on which you want to receive notifications. The service is only available to eligible ZIP Codes, so look for other options if your address is not in the system yet.
If you accidentally lost the USPS tracking number, double-check the receipts and e-mail confirmations to retrieve it. If you can't find it anywhere, your only other chance is to check if your order was sent through Priority Mail Express. If it was, you can ask for the tracking information in the nearest postal office.
If the status of your order has not been updated for a long time, don't look for alternative ways of tracking your shipment: this is probably due to the tracking issues USPS occasionally has. Contact customer support, and they will try to solve the problem.