I agree with Jeff, but a lot depends on the forcast and what you get. I carried an UL canoe into the St. Regis Area for a week...3 days of rain and thunderstorms unbroken for twentyfour hours and intermittitant the day before and after.
I took my sleeping tarp of course, so, I was warm and dry at night dispite 3 lightening storms. I also bring a 9'x11' UL tarp. This weighs about a pound. I set this up just over the fire, streatched back in a lean-to.
This is the overall extra's I bring:
Tarp: about 17oz
Extra socks: about 2oz
The extra 19oz is well worth it if the forecast is for rain.
Technique once you are done hiking (I was hiking between Mountain Pond, Slang Pond and Bessie Pond about 7mi of portage and 5 miles of paddling on this portage):
1) My clothes were soaked, of corse. My pack gear was still fairly dry, and my bag and sleeping clothes were dry. At one point it rained so hard, my hat was actually leaking...a rareity. Anything that can get wet is put into outside pockets. My food and bag (sleeping cloths) are inside the pack, in a liner bag. They stay dry. This included an extra pair of socks.
2) I immediatly found a dryer spot for my pack and sleeping tarp, set up the 2 tarps, and changed out of my wet socks into barefoot in my shoes (Teva sandals.)
3) I got enough firewood for the night...firewood was an esential because you need the extra heat to dry cloths. This means waking up periodically to feed the fire. I also set the bear line to save going out in the rain later.
4) I got water out of the pond, (filled water bottles and my pot.)
5) Got the fire going, painfull, but doable under the tarp. The rest of the wood acts as a seat. I strung a line from side to side(a couple sticks, tied at the center) and started changed cloths. I changed into my sleeping cloths (long johns) and slipped my sleeping jacket on.
(I carry a lighter bag, using a down jacket for those times I need extra warmth. The jacket does double duty around camp, but I am carefull to keep it dry.)
6) I laid out my pad and dried it off with a bandana. (The foam pad does not pick up moisture, but it does get wet in the outside pad pockets.) I laid out my bag and whatever I had for a pillow. I used my shoes later, inside the compression bag and the bag was turned inside out. I shook it out a bit for loft and folded it onto the pad to prevent it from picking up water from the ground.
7) After wringing out my cloths, they were hung under the tarp. They do not dry totally, but they dry pretty well. The pack gets hung up, as does my life vest.
8) Make supper, fill out my trail log, and just sit back and enjow the view and sounds or silence. About dark or so, I banked the fire down and added some fresh hardwood sticks. (It will burn slowly about one to two hours.)
9) I hung my jacket, changed into dry sleeping socks and went to sleep. I added my pack uner my feet, more to prevent the bag from getting soaked than anything else. On all the pine needles, I don't usually bring a ground cloth. After two hours of being under the tarp, the water was mostly drained away. Everything was still wet, but not really soaked.
10) In and out of bed all night, mostly as the rain or lightening woke me. Feeding the fire is one of the chores I need to do for rain...I HATE waking up on a 40F morning and putting on wet 40F cloths. At least they are somewhat warm, if not fully dry.
11) Swap cloths, and hang my sleeping bag while I make breakfast. I run down to get more water. Still raining... Made breakfast (getting the bear line, while I was at it.) Sit back and enjoy my coffee/cocoa.
12) Pack up my bag and cloths, douse any fire, scoop a pot of ash and scatter it into the woods, drop my sleeping tarp, cloths line, and pack everything except the lean-to. Then I drop my lean-to, rolling it tightly to squeeze some water out. I was ready to hike again by 0700.
Hanging the bag and sleeping cloths in the morning gives them a chance to dry out a bit. In summer or generally warm weather, I do not worry too much about dampness. I don't need that much insulation. You CAN hike in wet weather with only your sleeping tarp/tent, but this gets a bit uncomfortable. Clothing is wet. There is no way to dry it in a tent. It builds up a LOT of condensation if left inside or in a vestibule. Even a tarp can build up condensation in wet, muggy, conditions. Soaked clothing only gets your bag wet, if you try to dry it on your bag, too much moisture. In the west, it is rare to encounter this. In the North East or North West water and moisture is bad. Even my "dry" stuff is actually damp all the time. The upside is that you are generally fairly clean. At least part of the sweat and dirt is wrung out when you wring out your cloths. A wet bandana also makes a fair wash cloth. Socks are usually rotated through three pair. I make it a rule to keep one pair dry at all times. You might also need some extra food. A good hot supper and hot drink makes the weather seem better. A candy bar before bed helps keep you from getting cold. Not much of a problem in summer with temps dropping to the mid 40's by morning.
Enjoy it. The forest and hills have a different character in the rain. Don't worry about getting wet, provided you can sleep dry. Plan for the next day, though. At the end of three days of rain, even your bag will have water spots. Not a problem as long as the water is liquid. Freezing after a rain is painfull, though.