I have no problems with the hat shadow...
However, I always have a flash on my camera when I am shooting... I often use the smallish Canon 420EX for walk around shooting because of its small size relative to other hotshoe flashes. In fact, I purchased a second 420EX used on eBay to either use as a back up or to have mounted on my second camera. I use a Joe Demb Flash Diffuser Pro with these units. However, I will also use either the Mini Canon 270EX-ii or a Viltrox JY-610C when I just want a flash occasionally while walking around shooting.
Both the 270EX-ii and the Viltroxare tiny flashes and I don't even realize that they are on the camera. Both have advantages and disadvantages: the 270EX only has TTL capability but, it does have high speed sync (HSS) the Viltrox has TTL or manual capability but, doesn't have HSS. I can live without HSS because my Canon crop cameras can sync at up to 1/250 second and I am not doing specific portrait work with these small flashes so I don't need to shoot wide open. Both flashes are slave capable but, the Viltrox only costs about $25.00 USD on eBay...
http://www.ebay.com/itm/381673628960...%3AMEBIDX%3AIT Neither flash is heavy enough to cause the camera to be out of balance when I am carrying it on a neck strap.
I can use either flash with a little "shower-cap" diffuser if I desire.
http://www.ebay.com/itm/252476530511? Simply point the flash head up and place the diffuser over it. Does a pretty fair job._trksid=p2060353.m2749.l2649&ssPageName=STRK%3 AMEBIDX%3AIT
BTW: It's a wonderful feeling to share the bond with your dog. A friend of mine remarked recently that Sasha (my Jack Russell terrier) never takes her eyes off of me! I have six dogs sleeping in the office with me right now but, if I make the slightest move, Sasha opens her eyes to make sure I am still here.