jQuery update and wp_enqueue_script ?

Hello my theme (not official WP theme) is using this jQuery.

This jQuery is called from the header with this code:

<script src=”https://mysite.com/wp-content/themes/themename/js/jquery.min.js”></script&gt;

This file is jQuery 1.4.2 and I would like to use 1.8.3.

How should I proceed to update jQuery?

Would a simple file replacement work fine (replace jquery.min.js with a newly downloaded jQuery .js) or do I need to use the wp_enqueue_script?

Thanks for helping