FET to be controlled with less than 1V Vgs
You could use something like a MCH3383 which is a P-channel MOSFET that is guaranteed to have less than 0.5 ohm Rds(on) at 0.9V. 

Or perhaps the RYM002N05T2CL which is an N-channel part that has less than 9 ohms Rds(on) at 0.9V. 

There are others. Try going to a distributor website and do a parametric search, then read the datasheet thoroughly (the distributor sites often contain errors and are best used as pointers to the manufacturer's data). 



Another approach that is frequently used with a single-cell power supply is to use a boost regulator to increase the cell voltage to a more appropriate voltage such as 3.3V or 2.5V. Chips such the TI TLV61225 can do this (ON Semi has a number of them too). Keep in mind that the current drawn at the higher voltage will be reflected by a proportionally higher current from the battery (eg. 100uA at 3.3V will be more than 330uA from the battery at 1.0V).