Yes. Your new plan subscription will be started immediately and you will be charged the full amount for that plan less any unused days remaining under the old plan.
For example: if you had signed up for a monthly Simple plan on the 1st of January and decided to upgrade to a monthly Advanced plan on the 15th you would then be charged the full monthly price of the Advanced plan less the pro rata amount related to the remaining (unused) 17 days of your Simple plan. The next billing date for your new Advanced plan would be 16th of February.