What's the appropriate course of action if an airliner bounces 5-10 feet on touchdown?
Well, "5 to 10 feet" is not "slightly."  One or two feet is "slightly," and in that case you can maintain your landing attitude or pitch very slightly nose-down -- really a matter of the slightest change in pressure, not really a perceptible change in pitch -- and you'll land soon after that.

However, for a larger bounce, the concern, especially on the longer 737's, the -800 and -900 and -900ER, is a tail-strike. The tail skid protects the tail from a strike on takeoff, but on landing the geometry is different and the skid won't be what contacts the runway; it will be the fuselage.  And while tail strikes on takeoff are more common, it's the tail strikes on landing that are far more damaging &amp; expensive. So avoiding one of those is a major concern.  Thus, the usual guidance in the event of a bounced landing in the 737-800 is not to attempt to salvage the landing, but rather to go around.  You may touch down again during the go-around, and that's okay, but you're better off aborting the landing &amp; taking it around to try again, than to risk damaging the aircraft by trying to recover from a high bounce.

One thing that complicates recovering from a bounce is the fact that your spoilers may have extended when you touched down the first time -- and the last ten feet to the runway with spoilers fully extended &amp; power already at idle, won't look nearly the same as the usual last ten feet to the runway, with power coming off &amp; no spoilers. You'll settle fast, and hard, and you won't have the elevator authority you're used to for adjusting the sink rate.  You can pull back, but the change in attitude probably won't arrest the sink rate the way you think it will -- you'll just hit harder &amp; in a more nose-high attitude.  Thus the increased risk of hitting the tail.

On a shorter aircraft without the tail-strike concerns (say a 737-200,  -300, -500, -600, or -700), it might be possible to salvage the bounced landing, although 10 feet up is a pretty high bounce -- even if you had a lot of airspeed to play with and runway ahead of you, I'm not sure it would be wise trying to sort that situation out -- go around &amp; try again.