You can upgrade a server at any time by contacting your host. If you find your website slowing down or crashing frequently, you must immediately contact your host and upgrade to a higher resources option. The upgrades can be done within the same day. If you are on a shared hosting you can move to the VPS or dedicated hosting depending on the traffic and your budget.
If you are on a VPS, you can upgrade to the dedicated server. If you already are on a dedicated server and still need an upgrade, it means your business is in full flow and you must buy another dedicated machine. So, it is important you have a hoist that is available 24/7 to assist you in such matters. You may be finding traffic coming in more from one region and wish to increase your capacity, the technical team must be there to do the work. It is important for you to keep a check and you must order a new server a few days in advance so that there is in hindrance in smooth performance.
For upgrading from the shared hosting to VPS or dedicated servers you can uae the Upgrade form. You must pay for the difference in plan costs. You must not cancel your old account until the DNS are changed. You must order for a new machine online with your host by creating a ticket and send a request for free content transfer. In case you have any confusion, you can always talk to the support team over phone or chat.