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Starting with a prefab model of t-shirt I tried to scale it to fit better but the result isn't satisfying. Some parts of the models mesh become visible.

enter image description here

Is there any option or modifier that could add some spacing between body and cloth, or should I try to tweak loop-cuts manually?

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As satishgoda says in his answer.. the Shrinkwrap modifier is the way to go but it's best to not use a prefab in most cases. These are usually badly shaped or just have a different flow from the rest of your model. They can also prove to be very difficult to try and manipulate to conform to your object..

The best way to make clothing and fit it to your character is to select a part of the upper body in the shape of a shirt and duplicate it.

Here I have the upper body of a base mesh..

enter image description here

I next Tab into edit mode, select and duplicate the part of it that I would like his shirt to be and separate it from the body into its own piece with P > Selection. After this, I just scale it outward the tiniest bit with Alt + S to distance it a bit from the body then add a Shrinkwrap modifier to snap it back to the base mesh. This won't be perfect most times, (especially if you have bad topology) so it might require some tweaking on your part to adjust this..

enter image description here

After that, it's up to you now, you can join the meshes back together if you want to animate etc or you can apply the modifier and shape it a bit.. add pockets etc..

enter image description here

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    $\begingroup$ Those are some tight shirts, man $\endgroup$ – someonewithpc Jun 6 '15 at 18:00
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    $\begingroup$ After following all these steps, you can also try using a Cloth Simulation on the clothes to add some character that loose fitting cloths tend to have like some droopiness and creases. Once you run the animation, you should end up with a series of frames that all show different results. Pick the one you like the most and then 'Apply' the cloth modifier at that frame. :) $\endgroup$ – MarcClintDion Jul 1 '15 at 4:56
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ShrinkWrap modifier is an appropriate tool for this job.

You add a shrinkwrap modifier on the model of the tshirt and add the body as target. There is also an option to specify auxiliary target: like some other object that is parented to the body.

There are various options: one of it being "offset" that you asked in your question

  1. http://www.blender.org/manual/modifiers/deform/shrinkwrap.html
  2. http://www.blenderdiplom.com/index.php/en/tutorials/item/91-ressource-blender-modifiers-explained
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Another solution is to paint your own t-shirt on the body with the grease pencil tool and the Bsurfaces Add-on:

Bsurfaces Add-on

It's not only for retopology. You can paint different meshes on you body. Sometimes it's helpful to have another topology for your cloth than your body-mesh.

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