Perhaps, but not by much. Don’t get your hopes up.
From 1982 to 2014, the average annual increase for the experimental CPI-E was 2.9 percent, compared to 2.7 percent for the CPI-W. Using the CPI-E to determine the Social Security COLA would increase the expected average COLA by about 0.2 percentage points per year. The BLS acknowledges the current CPI does not “produce official estimates for the rate of inflation experienced by subgroups of the population, such as the elderly or the poor.” An index for the elderly could make a difference for millions of Social Security beneficiaries.
For every dollar you had in 1982 by 2014 you would have a $1.07 in 2014 compounding 2/10 of one percent. Plus using the CPI-E is not a steady additional increase. There have been years where the CPI-E has been lower than the CPI-W.
If you compound that .2% difference over the life expectancy starting at age 65 your $1.00 becomes $1.04.