Here is a manual on how to extend an expiring visa in Amman, Jordan. It is important to do so if you are staying longer than the visa states, because you would have to pay a fine when leaving the country without a valid visa.
(This is how it worked in April 2014. I cannot guarantee it will continue to do so in the future.)
While many travel guides tell you that you just need to go to the next police station, this is not correct. They will send you to the main police anyway.
This unit is located near Wadi Seer Bus Station, west of downtown, right at the street leading up to the third circle; see the map below.
Source: http://here.com/; amended: borderlandlevant
There you get a form you need to fill in and which has to be signed by you and the landlord/hotel/hostel you are staying. To make it easier for you, I put it up here (below) so that you can download it. The authorities do not offer this, why you would have to go there twice. As said above, fill in the form, make the landlord/hotel/hostel fill in their part and then go there.
Source: Jordan Police, Amman
When you print the form, also do copies of the first page of your passport and the page of the passport on which the visa is. If you don´t have a scanner/printer, this can be done at any of the countless shops in the city or in a small store right next to the police station; it costs close to nothing. I would advice you to also make a copy of the filled in form for your own records, before you hand it in.
Finally you need just a little patience, as bureaucracy is as it is and you will likely be sent from one desk to another and from one office to another just to get the thing inspected and stamped. Also be prepared that they again ask you where you are from or where in Amman you live, and so forth.
That´s it! You are welcome!
© Sigi Atteneder, 2014