What could be a near rail-to-rail opamp with very low voltage for this use?
IMO, your best bet is to do a parametric search on a decent distributor (Digikey has a pretty good setup).  Note that you will want to select rail to rail as a parameter, and then the power supply voltage range- read those carefully as they can be split up into single ended and dual supply types, and you can miss a good candidate such as NCS2001, NCV2001.  One important thing to note is that these opamps tend to have low gain-bandwidth products, so make sure the part you select has the chops for your microphone type (you didn't have that info on the schematic).  In the worst case you can divide up the gain stages (helps with oscillation problems too).

Most won't have a problem running with split supplies or single supplies, your choice, but you might also add an output coupling capacitor in case you connect something that has DC coupling on the input, which would drain your battery quickly.

The LT1494 would likely not work well due to its extremely low gain bandwidth product of 2.4KHz.