Tips for multiple-pronunciation characters (多音字)
To answer your specific question, it should be kept in mind that 尽 is a simplified character that merged both traditional characters 盡 (jìn) and 儘 (jǐn), which is why it appears to be a 多音字 in Simplified Chinese.  The difference between 盡 and 儘 is very subtle though.  This link talks about the difference as it applies in Cantonese, but I believe it applies in Mandarin as well:


  「儘」是「力求達到最大限度」，強調的是「力求」，至於能否達到「所求」，不得而知。「盡」則有「達到最大限度」的意思，強調達到。
  
  儘 is "strive to attain maximization", with the emphasis on "strive"; whether it can be attained is unknown.
  盡 on the other hand has the meaning "attain maximization", with the emphasis on attaining.