Is it considered a lie to inform colleagues and the manager of an untrue reason for requesting leave from work, based on the hadith "Accomplish your needs by discretion," in order to pursue another job that has not yet been secured?

It is legislated for a Muslim to conceal what, if concealed, would be a means to fulfilling his need, as stated in the Hadith: "Seek help in fulfilling your needs by concealing them, for every possessor of a blessing is envied." To avoid the forbidden lying when disclosing a reason, one can use ma'areedh (euphemisms/double entendres), which is speaking in a suggestive way intending one meaning, while the listener understands another. This is as narrated from Umar (may Allah be pleased with him): "Indeed, in ma'areedh there is room to avoid lying." However, stating a reason contrary to reality is forbidden lying.