Knowing how to re-tie any loose bowstring serving properly is an invaluable skill. It's a skill I use very day in the workshop when tying in peeps, re-tying in loose centre serving and end loops. All you'll need your Leatherman pliers and cigarette lighter to quickly and perfectly repair any loose serving in the field.
Have a look at this image - what you are looking at is a technique called a 'Bridge'. It's an ancient technique dating back to early seafaring days where rope work was finished off with all manner of splices and techniques depending on what was needed. The bridge is that arch of serving that you see looping over the strands that are wrapped backwards.
Imagine your centre serving has let go - start by unwinding the loose strand until you have a decent hand hold length of thread and the serving is being unwrapped off tight firm serving. If your loose end is too short - you'll find out soon enough and you'll end up undoing your work and starting off with a longer handhold of serving. Now make your bridge as you see in this picture... you're wrapping the loose strand back under the bridge towards the tight firm serving that's in good shape. The bridge always crosses to opposite side! You'll find out what this means in a moment. Basically you can make a bridge in two ways - one doesn't work - the other does. You know when it's proper because after you trap the loose working strand (see image #2, below); for each strand you trap and wind around the loose working thread - the bridge unwraps and you lose a wrap from the bridge. If you use 12 wraps under a serving bridge (like I do), you'll have 12 revolutions of the bridge to make until the loose strand is ready to pull through and snug up tight.
Pull by hand the loose working strand so the bridge gets pulled under your new serving and disappears. At this stage your serving looks great. Now tie an overhand knot on the loose end and grip that with your Leatherman and pull it even tighter. Trim this thread so only a 5mm tag remains and melt this with your lighter. Be careful doing this as you don't want to scorch your bow string - wind is you enemy out in the field when doing this final stage.