That is a great area. Cell service out there is very very spotty. Even GPS doesn't work in the depths of a canyon. I would personally take a hard copy map and description of the trail which should be easily found online. However FYI the Wave is a very popular trail and a permit is required which is done by lottery either in advance via internet or in person for a walk up lottery the day before the trip. I think they limit it to 20 permits per day. So make sure and check up on that if you are not aware of it. As such you probably wont have too much trouble following the trail as there is some signage and some rock cairns. On the walk up permit lottery I have had people tell me they walked right in with plenty of others there and they got one but they are only doing 10 in advance and 10 walk up and the walk up is for the following day. So if you walk up you may or may not get one just depends on your luck I guess.
some info for the lottery and the hike
https://www.thewave.info/index.html
Since it is a lottery and you may or may not get a permit having a plan B would be a good idea. Some other hikes that are not too far away and while not the Wave they are awesome and some may say better but that would totally depend on your point of view
Buckskin Gulch is right there as well. One of the Grand Daddy's of slot canyons and not technical though it is a long haul from end to end (longest in the US) but you can hike in however far and turnaround and go back the same way instead of going all the way down to the Paria river and then up it to the White House trail head there which iirc is pushing 20 miles and you still need to get back to the starting point to get your vehicle so if you dont have two vehicles or a shuttle dont try to go end to end.
http://climb-utah.com/Escalante/buckskin.htm
Some others such as Round Valley Draw an excellent slot canyon that does require a bit of scrambling but nothing serious, Bull Valley Gorge same thing. Also Willis Creek and I think Sheep Creek. A piece of webbing or rope 20-30 feet long would be good to take along just in case for help if needed getting up, over and down a choke stone or short pour off. Both of those are up off Cottonwood Canyon Rd which the southern terminus of that road is on the highway not far east from House Rock Road where the Wave is located. A couple others in Page are Antelope Canyon upper and lower. Waterholes Canyon is another in Page. A quick web search will get you info on those. The Page hikes are on the Navajo Reservation and require permits from them which can be gotten at some trail heads or their office just outside of Page last item I was there anyway.
Probably too far but Hole In The Rock Road outside of Escalante is teeming with slots and great hikes.
