How about using a cheap eBay DC Dc converter board to drop the second side 18VDC or so to around 8V, which will give you enough headroom for the linear regulator. Saves power, will give a higher current output on the 6V side and is cheap ( $2 each) so you can buy a spare one or two, or just order 2 of the buck converter modules with a built in controller and led display, and have a power supply with similar capability but with 3A output capacity over the full range. Slightly more noise on the output, but you can add extra LC filtering on the output to reduce that a little.
Probably cheaper to buy the switchers on fleabay than the LM338 anyway, plus you need a smaller heatsink as well.