Haystack's email signatures are smart - when you update your digital business card or the card's template, the signature updates itself automatically - no need to do anything else.
The reason you might still see the old signature in your email client is because your device may have 'cached' the old signature into its memory, so while the signature has already updated for everyone else (including in emails you've already sent prior to the change), your computer may still be showing you the old version.
Not doing anything for a few hours would give your email client and/or device the chance to update its memory, and you will see the update reflected on your device soon enough (normally within hours, but can take up to a couple of weeks in some cases).
If you want to test it, send an email to a colleague, or your personal email address, and check the email signature as a recipient. You'll see that other people get the updated version.
PS. If the recipients of your emails receive your old signature - please contact our support for assistance.