I can compute some other values:
Based on the True Money Supply (i.e. the increase in the quantity of dollars), I get an answer of about $632:
http://mises.org/markets.asp#More%20on%20TMS
Based on the price of gold, I get $655.
Based on the price of crude oil, I get $642.
In all of these cases, the value is much higher than the $293 you gave.
Of course, there is no objective means to compare someone's valuation of a dollar today with someone's valuation of it in 1996. But I surely wouldn't use the BLS - data as they intentionally strive to give false inflation figures for political goals. The three latter values, $632 - $655, seem most trustworthy IMHO.