When preparing an application for a position, it goes without saying that everyone wants to make the strongest possible case for themselves. Hence, it seems that the best strategy would be to wait until the last possible moment - just in case in the meantime an acceptance letter comes from that journal where you submitted the paper half a year ago or your latest project reaches a stage when it can be posted on arXiv.
However, I am wondering if this is reasonable. Presumably, whoever is assessing the applications has internet access and can easily check if there are any updates (say, by visiting the personal webpage or googling the applicant's name). Since application procedures tend to be somewhat lengthy, I would also imagine that hiring committees are well aware that the applicants accomplishments can increase in the meantime.
[The position I have in mind is maths postdoc, but I welcome a broader perspective.]