When I visit op shops I look for second hand Xmas CDs -There seem to be plenty to choose from. Occasionally I find one with some original or different songs. These I collect for my own Xmas mix.
Any songs from the Bing Crosby songbook are too cliched to be included.
My favourite song is Tim Minchin's White Wine in the Sun. It expresses the atheist's Xmas perspective. Kate Miller-Heidke sings a great version on the album Spirit of Christmas 2010.
My favourite album of unusual Xmas songs is the Best Ever Christmas Mix - from EMI 2004 - 724387476820