> Adding a pull-up resistor to the output of the LM358 ..
> Say 1k ~ 4k7 or so

Thanks for the suggestion. I did some tests

Without resistor

Supply +5.4 / 0V  o/p swing 0 - 4.15V
Supply  +5.4V / -5.3V o/p swing 0 - 4.14V

With resistor (various 1k - 10k)

Supply +5.4V / 0V o/p swing 0.915 - 4.80V
Supply +5.4V -5.3V o/p swing 0 - 4.80V

So a resistor helps, but not quite enough. I suppose you
could use 5.4V Vcc with 4/5 gain and have 4.096V out
= 5.120V in as an alternative to a higher Vcc

> Without it they are awful - can only drive to within 1.5V
> of +V (from my memory)

According to the d/s that's right. In practice a little closer

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