Chris,
The problem is that the LM358 is not a "rail to rail" op amp as the LMC 662 is.
If you look at the data sheet for the LM358, The maximum output for a 5 Volt supply is typically 3.5 Volt. This is why you can't make it go higher than 3.4 V.
Also, if you look at Figure 1 in the data sheet (I have the Motorola DS) you will notice that for a 5 Volt supply, the input voltage range is only 4 Volt.
So you have 2 options:-
1. Use a higher supply voltage or
2. Find another rail to rail op amp.
Also, I don't see why you need 2 op amps. If you want a gain of 5, then you can do it with one amp.
Remove the second op amp and change the feedback resistor from 10k to 50k (ie. two 100k resistors in parallel or a 51k) and replace the 10k between the + input and gnd with a 50k.
This will make the differential gain = 5.