Rather then giving you the link I'll post the step sequence so you'll know how to do it yourself:
Go to
www.uscis.gov
Click on Services and Benefits top left corner of page
Click on Accelerated Processing bottom of page
Click on USCIS Expedite Criteria right corner of page
Read the expedite guidelines and compose a memo arguing why your case fits within the criteria described above.
Search within the Service Center List, based on the state you live in, whether you have to file in Vermont, Nebraska, Texas or California. There are just four Service Centers.
Call 1-800-375-5283 and ask to speak with an
immigration officer. The employees who answer that phone in that first place are not immigration officers and often have no clue what they're talking about beyond the general information available to the public on the website. They sometimes give out the wrong information. Ask to be connected with an immigration officer.
Tell the officer what state you live in, confirm that the Service Center you located is correct and ask them how to file I-102 expedited at your respective Service Center. Most likely he will tell you to file by mail, wait a couple of weeks for the I-102 notice of action to arrive in the mail, and then once you have a case number, fax a expedite request to the Service Center to which they are obligated to respond within five business days. The officer will provide the fax number for the Service Center if you ask him nicely. Of course, responding doesn't mean they're going to reply to your fax, but that they're going to take some prompt action in your case, i.e approve or deny the expedite. Wait a week after your fax, then type in your case status online or call the 1-800 number. Hopefully your expedited request will have been approved.
I filed I-102 in 2005 and it took 3 weeks to get a replacement I-94 effectively. Otherwise it takes months. It costs nothing to file expedite. The $1,000 fee is just for accelerated processing, which as you noted yourself is something else.
As for the POE stamp on the passport, it is useful to prove legal entry in the US, but in most cases that stamp doesn't tell your interviewer the authorized length of stay, nor provide your Arrival/Departure Number. This is why it essential to have your original I-94 or a replacement.