$('#start').timepicker({
  change: function(time) {
    timepicker.format(time);
    $('#end').prop('disabled', false).prop('readonly', false);
    let startTime = $(this).timepicker('getTime');
    startTime.setHours(startTime.getHours() + 1);
    $('#end').timepicker('startTime', startTime);
  }
});

$('#end').timepicker({
  timeFormat: 'h:i A',
  interval: 60,
  minTime: '01:00',
  maxTime: '23:00',
  dynamic: false,
  dropdown: true,
  scrollbar: true,
  change: function(time) {}
});

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.1/jquery.min.js"></script>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/jquery-timepicker/1.14.1/jquery.timepicker.min.css" integrity="sha512-WlaNl0+Upj44uL9cq9cgIWSobsjEOD1H7GK1Ny1gmwl43sO0QAUxVpvX2x+5iQz/C60J3+bM7V07aC/CNWt/Yw==" crossorigin="anonymous" referrerpolicy="no-referrer" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery-timepicker/1.14.1/jquery.timepicker.js" integrity="sha512-mgOX10VJOglmFMwk8ErnhCLJDofOiZh+YJGzD30SsGhCqRQrx+dpTpnSu4XaB/Ob0OUDDzYb6ORlRjjHk5AdnQ==" crossorigin="anonymous" referrerpolicy="no-referrer"></script>

<label for="start">Start Time:</label>
<input type="text" id="start" placeholder="select time" autocomplete="on" name="start" required>

<label for="end">End Time:</label>
<input type="text" id="end" placeholder="select time" autocomplete="on" name="end" required disabled>