I have a similar problem! I love Meg, but dislike the natural "origin name" Margaret (and Megan's a bit off, even if I/you did like it, because it's pronounced more like MAY-gun).
I like the name Megara, but most will only associate it with the Disney princess as opposed to the actual Greek mythological figure (even in the myth, she doesn't do much, and one version has it she's killed by her husband. Yikes).
I've heard of people using Magnolia, Marguerite, Marigold and Magdalena (and all its variations) as well.
There's also the unusual-sounding Megane, which is just different enough from plain old Megan you might give it a try.
The Japanese name Megumi has a lovely meaning ("blessing" or "blessed beauty") but is very unusual.
There's also the old standby of using a combo with an M first name and G middle, a few E's somewhere in between to try and get to Meg.
Names with Meg:
Mega -- mild, gentle, peaceful
Megara -- pearl
Meghana -- thunder
Megiddo -- from the army's camp
Megumi -- blessed beauty
Names with Mag (I know some people take Mag names and use the nickname Meg)
Magalie -- It is known as the title of a Provençal folk song, and could be derived as a Provençal form of Margaret. You could also just change it to Megalie which I actually think is cute
Magdalena -- of Magdala
Magena -- the coming moon
Magna -- great
Names with Marg (since Margaret is the origin name of Meg)
Margana -- one who seeks the truth
Margi -- direction; marjoram herb; pearl
Margot/Margo -- pearl
Marguerite -- pearl
Names that start with ME with a G middle name (there's a lot of combos out there so I'll just make examples)
Magalie is cute! I have never really like Marguerite because it reminds me of margarita. I used to love Margot so much, but I'm not too sure on it now. Audrey Charlotte & Margot just doesn't seem right. Margery/Marjorie is ok, but seems a bit dated. I kind of like Mara or Mary (Mara is a family name) but I am not a huge fan of G names. An Allera or Morgana could go by Meg, but both are much to heavy for me. Man I'm picky!