The Basics 
This site endorses the practice of - Folding down the best hand over dollar value.
Making adjustments for dollar values between players when raising occurs, is highly important to take into consideration.
I've folded down many a winning hand in middle and late bet rounds to re-raisers because I haven't adjusted the value of my dollar to the dollar value another player may place on the cards in their own hand. It can be a challenge to push through top pairs vs. other two-pair if I'm not keen to the value another player places on particular cards in play.
We all pretty much know by now that getting a feel for the players around the table is critical, and the sooner the better. An adjustment that needs to be managed immediately is observing the bet value that other players place on particular cards. Especially opening single pairs and two pair on later streets.
At the door, a pair of hidden or split Jacks may very well be worth completing the bring-in bet, especially if there are no Aces showing. However, for me, it does not warrant re-raising a player that has made a complete bet under average circumstances. For example, a door card 8 completed the bring-in bet in early position. I am in late position with my pair of Jacks. I would personally just call and not re-raise over the 8’s completed bet, whether or not Aces are showing at the door or if I have an Ace in my hand.
To another player, re-raising a completed bet with a Jack showing at the door may be a tell as to the dollar value that they place on that particular hand, letting you know that they will raise when they have a pair of face cards to open with. Adjustments can be then made when participating against a player so that you are not continuously raised out of a hand that may just be the best one. Some players will three bet when they have any high pairs with live outs to three-of-a-kind. Simply keep an eye on players who are three-betting at the door and what they are showing down with. Someone three-betting a pair of 8's to make you fold down your split pair of tens to open may be costing you.
In some stud 8 games at the door, lows can be likely to raise and re-raise with a draw to a low, perhaps disguising a flush draw, or straight draw as well.
It’s not to say that re-raising over the 8 with a pair of Jacks would not be correct under favorable circumstances. What I am pointing out is that in general, while a pair of Jacks in my hand doesn’t warrant a re-raise by my personal strategy for a full table or a multi-way pot to open, doesn’t mean that it is the same for another player’s strategy. Re-raisers may often believe that the hand they are carrying at present is truly the best hand. A raise of $5.00 means differently from one person to another.
So just as an FYI, a reminder, or however you want to view it, keep in mind
to be aware of the dollar values players put on their hands and include it into
your evaluations so that you aren't folding down the winning hand out of fear
when there may be nothing to fear at all.
