My first guess was "death". I spend a while just trying to fit different letters in different places without any proper result. I eventually just started writing out all possible options, using an * as the missing letter:
E*HAD
E*HDA
E*DHA
ED*HA
EH*DA
EA*HD
EH*AD
EHD*A
EDH*A
EAH*D
EDHA*
EHDA*
EADH*
EAHD*
A*HDE
A*HED
A*DHE
A*EHD
AD*HE
AH*DE
AH*ED
AHE*D
AHD*E
ADH*E
ADHE*
AHDE*
ADEH*
AHED*
I wrote down all the options with the E and the A as the first letter and only while doing a quick second pass over the A options did I notice that AHEAD would be an option.