Should Ph.D. students interested with research remain as research assistant until they find a research position or just find any job?
There is a delicate balance here: in certain situations, it may make sense to be strategic about your graduation date, especially in order to align yourself better with hiring cycles.  If you start stretching out your thesis work significantly, however, it will likely be noticeable and not to your benefit.

More to the point, any such delaying tactic will almost certainly require the consent and participation of your advisor in order to be effective: otherwise, you are likely to be pissing them off and making your chances worse rather than better.  Have a discussion with your advisor to see what their assessment of the tactical situation is, and their recommendation.  

Finally, with your advisor's consent there is another option worth considering: a "transitional postdoc" in your current location.  In many institutions and/or fields (especially private institutions or expensive lab fields), the cost of a postdoc is not significantly different than a graduate student, since the higher salary is offset by the lack of tuition or made less significant by laboratory materials costs.  For a transitional postdoc, you graduate but you stay briefly as a postdoc (no more than a year or so) while you continuing working in your more independent role to increase your profile and position yourself for an external postdoc or long-term hire elsewhere.