Flight Instructor reported an NMAC during departure after the tower had closed for the day.

Narrative

During left crosswind turn on at PQL; my aircraft nearly got inside of Aircraft Y traffic established on the downwind for 17 during our climb out. Shortly after the Aircraft Y called 'downwind 17'; my student visually cleared the aircraft and made the crosswind call. Upon rolling out on downwind; Aircraft Y queried our position; to which we replied 'downwind 17'. CFI then noticed Aircraft Y traffic slightly ahead and abeam at roughly 5 wingspans distance. CFI took controls and sequenced behind the other aircraft to give way.CFI did not see the traffic prior to the crosswind turn. There is potential they could have been masked by our wing or other part of cockpit. Aircraft was equipped with Stratus and CFI EKB (Electronic Kneeboard) was connected; but admittedly was focused primarily outside VFR scan. Given the length of time between the other aircraft's 'downwind' call and our crosswind call; it seemed proper deconfliction and interval had been established between other traffic in the pattern. Despite best efforts to visually confirm downwind aircraft's position; we were unable to locate them and in our judgement we were clear to begin crosswind turn. As the CFI; I should have taken a moment to reference ADSB to ensure clearance before allowing the student to turn. Luckily this ended in a non event but nonetheless has served as a sobering learning opportunity for both my student and myself. Of note; PQL is usually a towered Class D but was inactive at the time and reverted to Class E airspace / CTAF frequency.