Is it wrong to write 5.58 followed by one third for instance? The number I am imagining is 5.58333333333. I know I could put a dot over the first 3 and omit the rest.
Yes, that's the crucial point. In general it is not good practice to combine decimals and fractions. In this particular case the reader might think you meant $\displaystyle 5.58+ \frac{1}{3}= 5.913333\cdot\cdot\cdot$. Perhaps it would be best to use just the fraction: $\displaystyle 5\frac{525}{900}$ or even $\displaystyle \frac{5025}{900}$.
I would not write $\displaystyle 5.58\frac{1}{3}$, but if it's convenient you can define notation however you want, as long as you do so clearly.
Options that are more standard:
$\displaystyle 5.58+\frac{1}{300}$
$\displaystyle 5.58\overline{3}$
$\displaystyle 5.58(3)$
$\displaystyle \frac{67}{12}$
$\displaystyle 5.58333\dots$