How should I decide how many decimal places to use in a confidence interval? My T table look up is to 3 decimal places ( or less is I am estimating for n=39 say)
Yet my margin of error can be calculated to as many decimal places as I choose.
How should I decide how many decimal places to use in the confidence interval ?
$\bar{x} \pm t_{\frac{\alpha}{2},n-1} \frac{S}{\sqrt{n}}$

It's usually more related to the number of significant figures than decimal places.
https://en.wikipedia.org/wiki/Significant_figures
A table that shows a probability to say three decimal places (common with z-tables) may have fewer than three significant figures in the tail. Many tables also only give the t-values to about 3 figures (perhaps slightly less if anything, since the t's/z's typically only range from 0 to about 3, though you may gain some of that back with suitable use of interpolation).
In your case your linked table gives t-values for specified tail areas; which is fine, the tabulated values should be accurate to 3 or 4 figures. [Unless, that is, you're after an accurate p-value, in which case, it's not much good to you. I should also add a note of caution; it doesn't hurt to double check multiple values in the table because I've found a few tables with errors.]
In the final values you report, you should generally avoid giving more significant figures than the least accurate one in the calculations performed (so if the data was available to 5 significant figures and the table to 3 significant figures, generally you wouldn't report your final interval ends to more than 3 significant figures since the answers are likely no more accurate than that, and sometimes possibly less).
However, intermediate calculations should be carried out to more figures, since premature rounding off can result in dramatic loss of accuracy. I see this particular error made very, very often and sometimes the loss of accuracy from premature rounding is substantial -- I have seen reported answers that had only about a single digit of accuracy.
In general, where you have any option, don't use tables at all, but use software calculations that typically tend to have many more figures of accuracy, though some may be based on approximations with only a few figures of accuracy, so sometimes you need to investigate what calculation was used for some implementations. Indeed some packages have not updated some of their test calculations since the late 1960s even though better approximations may have been around for decades. Sometimes the fact that the calculations only work up to some very small sample size like n=50 - with the limit coming from some early paper - is the only obvious clue that those responsible have not looked for any more recent calculations.
