Does anyone know of any reputable literary magazines which publish this type of short story? I would prefer a magazine that takes free submissions.
You should check From short fiction to a novel– PsicofreniaAug 26, 2013 at 9:49
Please have a look at these links. They can be of help :
This website contains all the literary magazines in which you can publish your short fiction stories.
This website is a long hub of literary activity that supports aspiring writers. It is a non-profit organisation, a registered charity, with the remit of promoting writers and literature at home and abroad.
I think this will help.
Once you've used lists like these to identify magazines to submit to, you can also look at the magazines' submission guidelines, and the fiction that they have already published. These will help you narrow down your list of markets even further.