If you want to get site indexed quickly then yes and use the "fetch site" in Webmaster Tools. Also pinging the site and submitting its RSS feed using free submitter sites can also help. I have had sites indexed in less than an hour doing this. TBH you should always submit the sitemap to google and bing for any site.
A new sitemap means you have to submit another one or you want to replace the previous one. If you have added pages to your website then create a new sitemap and upload it.