I think the answer is somewhat dependent on the type of products you are selling, who you are selling to and where they are located.

If you are selling digital products, ensure that after the customer pays they get an email link or are moved directly to the download page. Make it as automated as possible.

If you are selling physical items to local people (clothing, accessories, etc.) use either part time workers or freelance / contractors to hand deliver items via car, bike or walking depending on the location. It will be faster and probably less expensive than sending through traditional post. Also, it shows a touch of personality with hand delivery.

If you are selling physical items to people at a distance (across the country or in another country) you may not have many options for cutting out traditional delivery. However, you could look into Drop Shipping, which I admittedly know almost nothing about. But I know it can be a way to shorten delivery time for customers.

One last idea for reaching customers at a distance, would be partner with another business and try to send your shipments out together (if you can get a bulk rate or discount). I'm not sure that is available, but it is at least worth considering.