It is right that you said "-기도 하다" means "I also do...". But actually... I think it is right to see the verb, "-ㄹ 것 같기도 하다" just as one verb. You do not need to bring "-기도 하다" here.
So basically, "-ㄹ 것 같기도 하다" and "-ㄹ 것 같다" are similar in meaning. But still, we can distinguish these two verbs. The first one kind of implies that the speaker did not think that it will happen, but now he thinks it will happen(by seeing some clues... or he could just have changed his prediction).
For example,
I think it will snow tomorrow = It seems it will snow tomorrow = 내일 눈이 올 것 같기도 해요(as now I am seeing the dark clouds / as I have just listened you say so / as I have chanced my prediction) = 내일 눈이 올 것 같아요