How to find Exosuit upgrade locations (drop pods)

To upgrade your Exosuit, you have one of three options to choose from:

Signal Booster
In order to locate drop pods, you need to build a Signal Booster. To build this, simply up on D-Pad for PS4 and X1 or Z for PC and scroll through the list until it appears. Once you have built the Signal Booster, interacting with it will prompt you to insert a drop pod coordinate data. After the chip has been inserted, the option to scan for a drop pod^1|2 will appear.

Exocraft
You also have the option to search for drop pods using an Exocraft with the Signal Booster Upgrade Sigma tech installed. If you have this tech installed, simply press the scan button or open up the quick menu to scan for drop pods. It is likely you will also need the drop pod coordinate data, but at the time of writing, I am not 100% sure

Space Station
While your question indicates you would rather not continue to warp between systems, this option is still available to purchase for units if you do not have the proper resources to scavenge the planets. However, if you do choose this route, be sure to avoid uncharted star system. These systems lack space stations and you will not be able to acquire the Exosuit upgrade.

After your hunting is complete and you have located the Exosuit upgrade (either on the plant, moon, or spacespace) it will provide of one three different types of inventory upgrades

1. General, which is standard inventory space
2. Cargo, which is a separate inventory space dedicated to stacking products
3. Tech, which is a separate inventory space decided to installing technology.

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Additional Sources (provided by chat discussion)

• GameFaqs¹
• Reddit²

I actually did this a much quicker way than visiting "Shelters". I actually just flew over the planets surface and searched for the drop pods myself, because it was much more quicker and efficient for me than having to visit shelters the whole time.