Does お好きに僕に連絡かけてください sound natural, and does it make sense as a sentence? If not, what would a more natural alternative be? In particular, I'm unsure about お好きに and ください in the same sentence.
Correct me if I'm wrong, but I think you're trying to say something like
Please feel free to contact me again.
Maybe? From your choice of 僕, I'm guessing this is not necessarily an overly formal context.
お好きに is not used like that. (Did you get this from お好きにどうぞ?) Depending on context and tone, もしよかったら or お気軽に or いつでも might be usable, although I don't know why you need it here in the first place. It's not like また連絡してくださいね would be interpreted as an obligation. (That would be お返事お待ちしております。よろしくお願いします。)
僕に is superfluous, especially with 連絡する (see below). Who else are they going to contact?
連絡かける. It's 電話をかける and 連絡する.
Without context this is obviously a shot in the dark, but...