Here are some notes on how I manage contacts in the contact puzzles. To get the initial placement of the protein, run Band All Contacts Adaptive (BACA) with contact weight >.9, band to beta carbons, and band strength 1 for both made and unmade contacts. This marks the most important contacts. Use those bands as a guide while moving pieces of the protein around, trying to make all those bands be as short as possible. Keep structures frozen, loops unfrozen, disable the cutpoints (but place a band on each cutpoint), and wiggle at low ci (I use .2) to let the bands move the structs into better position.
Alternate this with hand moves until you are happy that the bands are as short as they can be (that is, the contacts they mark are as close as can be). Remember when shaking to disable the bands so the sidechains are free to move. Feel free to cut and bend structs as needed. If you make major changes, re-run BACA with above settings to get new lengths for the bands. Use Band Breather to alternately expand and contract the bands a few times to settle the (still frozen) pieces. Unfreeze and wiggle (still low ci) - your contact score should improve now that the structs can bend.
At this point I delete the bands and use BACA to create new ones, weight >.8, band beta carbons, band strength 1*weight (Madde's version of the script lets you set the band strength as a multiple of the contact weight). Wiggle again unfrozen. Do more hand work if you want (there are more contacts to see now, so you may want to adjust things), alternate with wiggles, make new BACA bands any time they need new lengths or you want additional contacts marked with bands (in this case lower the minimum weight). Close cuts when ready.
To get your final contact bands (you will keep these for the rest of the puzzle!), delete existing bands, run BACA with weight >0 (this will band all contacts), band beta carbons (I suggest all, for reasons that will become clear later, but you can switch to hydrophobes only if you prefer) and strength .8*weight. Use Band Pruner to remove bands >13 (or thereabouts) long. Wiggle (still at low ci) until stable. Save your work!
At this point I run TDRW with long segments (I use 9-10) at low ci (I use .2) for a long time (generally until it covers the entire protein) with those same contact bands on. After the .2 ci run, I run TDRW again at .5 ci with medium segments (5-6), still with those same bands. Your contact score will fall when you first go to a higher ci, but keeping the bands on during scripts will bring it at least part way back up again. I have found that replacing the contact bands at any point lets the contact score fall.
The only bander script I use is Comp Band Keeper - although others could be modified to keep the existing contact bands in place and use them for the non-compressing wiggles. I also do a short seg (2-4) run of TDRW while still on auto. Any script that keeps the contact bands and uses them instead of any non-banded wiggle is fair game (I use only TDRW and Comp Band Keeper).
When it's time to go to medium wiggle, the contact score tends to go down, and at ~50 points per contact this can be deadly, so I put all my energy into protecting the made contacts. This is like playing whack-a-mole. On the contact map, I select all contacts, then (by hand) unselect all the ones that are not made (i.e. the empty green squares). This means every made contact appears in green on the protein, and there are no red contacts. (If you used beta carbons for only some of the BACA bands, the green contact lines where you didn't use beta carbons may be hidden by their bands, and you won't be able to see them). Disable (don't delete) the contact bands (still the ones you made way back during low CI!) and start wiggling on medium. As soon as you see a contact turn red, stop and undo the wiggle. Enable that contact band and set its strength back to default (1). Wiggle again until a different contact turns red, stop and undo, and enable and strengthen its band. Keep doing this.
Some of the fragile contacts will flip back and forth between green and red as the wiggle goes on; as long as they are settling on green this is ok. If any are settling on red, make their bands stronger (1.25, or if that is not enough, 1.5). Sometimes strengthening one band will make a nearby contact break - just keep enabling and strengthening bands until they are all stable. Eventually you will get all the way through the wiggle without losing any contact score. If you happen to gain any new contacts (rare but it can happen), be sure to go select the new contact in the contact map (a black dot in a green square that is not yet selected), and enable and strengthen its band to protect it.
At this point I run Cut & Wiggle, but I watch to see if any contacts keep breaking - if so I stop the script and reinforce those contacts, wiggle to stable, and restart script. For end game, any script that uses the enabled bands for wiggles is fair game - if you ever wiggle without them, some of those fragile contacts will break!
Hopefully this will give you some ideas you can use to manage contacts. Feel free to add your own strategies!