Every pin is different. The first step is to try an guess how the boat became pinned and the easiest way to jostle the boat. If safe enough, Get a person or two out to the boat. Pull on the ends to see if the boat can be moved. See if the boat can be rotated in a way that migrates the cockpit downstream. Where possible, determine how best to use the current to help rather than hinder your efforts.
Now that the analysis stage is completed, set the tag and hauling lines in advantageous positions. If rotating, perhaps you can attach to a pin bar close to the cockpit and use gravity and current to your advantage. If the boat can be moved forward or backwards, attach to that end and pull. Usually you only need to move the boat 12" and the current will do the rest for you. If stuck in a sieve, chances are you will need to pull back out the same way it went in - that is from upstream. If caught in an undercut that appears to be clean with lots of flow coming out, you may need to pop the air bags. In this situation, you will need to be exceptionally careful to avoid the same fate as the boat.
On dam released rivers like the Savage, consider tying off and wait for the water to drop. This is a common occurrence on this river unfortunately.
Paddles can be a real challenge to rescue, especially when lodged in a sieve. This is just another reason to carry a breakdown paddle or hand paddles. It may be possible to pull and twist but not always.
Use manual pulling and jostling where possible. Next, pull from a safe distance with lots of helpers. If you are really shorthanded, consider mechanical advantage systems like the C-Drag or Z-Drag.