It's easy to add your MapKnitter exported map as a background layer when editing OpenStreetMap in Potlatch 2. In this example I'm using a map of Mestia, a city in the Svaneti region of Georgia.

Potlatch 2

With help from Andreas Trawoeger from the OSM-talk list, we set up a "deep link" so that simply by clicking Use your map to contribute to OpenStreetMap underneath the Export formats area, you can open your map directly in OpenStreetMap's Potlatch 2 editor by clicking "Use your map to contribute to OpenStreetMap":

Note: this only works for maps which have been released under open licenses, such as CC-BY or public domain.

iD Editor

By clicking the Edit tab on openstreetmap.org, you can choose the iD editor, then click the background tab to add a custom layer. The URL you paste in will be formatted like this: https://mapknitter.org/tms/2011-7-31-brooklyn-gowanus-vis/alt/{z}/{x}/{y}.png

Also, you can use the openstreetmap.us installation of iD editor, and provide a URL similar to the one below:

http://openstreetmap.us/iD/release/#background=custom:https://mapknitter.org/tms/2011-7-31-brooklyn-gowanus-vis/alt/{z}/{x}/{y}.png&map=16.56/-73.98934/40.67737

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Old technique (no longer necessary)

First, open the "backgrounds" menu and choose "Add":

Then, add a new background layer, using your map's "OSM-style TMS" link, with added z,x,y like this:

https://mapknitter.org/tms/mestia/alt/$z/$x/$y.png

Where "mestia" is your map's name. See screenshot:

You can turn off "dimming" to see the map more clearly:

Zoom in and see how well your balloon map fits existing data. Then trace to add more features: