Yes, a traditional 55cm is small for a 6'2" person. Adjustments to make it fit you better can be made with stem, handlebar, and seat post. For example, you may want to get a longer stem and run it higher up; handlebars have different reach and drop measurements for fine tuning fit; a setback seat post can let you slide the saddle back a bit further.
Which combination of this things will work for you is almost impossible to say, but probably looking at a longer stem to get more space is a good place to start. Pics of your bike and, ideally, you on the bike, will help dial in suggestions.