Yes, go ahead and add more chlorine.
Also, your ph is too low. Any readings below 7.0 are acidic and can damage your pool. Since most testers only register down to 6.8, your ph could actually be much lower than that. Add some 20 Mule Team Borax (from laundry aisle at Walmart) slowly into the skimmer, breaking up any clumps. This will raise your ph. I would probably start with 1/3 of the box. Then, wait 3 or 4 hours, and retest ph. Then, if it is still below 7.2, add some more. It may take several doses of Borax to bring you into range. (Anywhere between 7.2-7.8 is good, but while you are fighting algae, on the lower end of the range is best.)
Ignore the advice to not do anything but chlorine right now. It is CRITICAL that you get that ph level up ASAP. It is ok to add Borax and also chlorine now. But, don't add anything but those two things for now.
By the way, if the pool store tests your water until you get a good kit, that is ok, but don't buy anything unless you have advice from here that you need it. Pool stores often sell you stuff you don't need -- cuz they are trying to make money, of course!
Keep us posted how things are going.