<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title>Document</title>
</head>
<body>
<table>
<tr>
<td id="start_1">12.09.2017</td>
<td>
<button id="button_apply_1" class="schedule__apply">Записаться
</button>
</td>
</tr>
<tr>
<td id="start_2">13.09.2017</td>
<td>
<button id="button_apply_2" class="schedule__apply">Записаться
</button>
</td>
</tr>
</table>
<form class="overlay-form">
<input type="hidden" name="data_start" value="">
<input name="name" placeholder="Ф.И.О." required="true" type="text"/>
<input name="tel" type="tel" placeholder="ТЕЛЕФОН" required="true"/>
<button class="overlay-form__apply">Записаться</button>
</form>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.2.1/jquery.min.js"></script>
<script>
$(function(){
$('.schedule__apply').click(function(){
$('input[name="data_start"]').val($(this).parent().prev().text());
});
});
</script>
</body>
</html>
так?