Both "tässä" and "tässähän" are equally as much "right here" since they both mean "precisely here". Without context, it's a bit difficult to say how the han/hän suffix should be translated, but in any case it doesn't really chance the location or make it more "here", so to speak. It's more like "surely", "right?", "you know?" etc. i.e. it modifies the tone of the whole sentence while simultaneously emphasising that the word it is attached to is the point of e.g. some confusion etc.
"Sinähän sanoit siitä minulle." - It was YOU who told me about it (,remember?)
"Tässähän se on." - It is right HERE (, see?)
So, using the han/hän can indicate that you want some confirmation on your thoughts (being correct) or that the person you are talking to knows the same things about the situation as you do, etc.