In photoshop, you will need to convert the DPI of the image, without changing its image scale. You do this by going to
Image > Image Size... and then within the dialog that appears, uncheck
Resample and change
Resolution to 600.
Now, it is important to note, once you have changed the resolution (DPI, or "PPI" as they show, Pixels/Inch)...you need to check the width and height in INCHES. Make sure that your image actually supports being printed at the desired resolution at the necessary physical dimensions!! This is very important, as if you do not check this, it is entirely possible that your image will print too small. If it currently prints at the correct PHYSICAL dimensions at 300dpi, then once you change the dpi to 600 the physical dimensions will cover ONE QUARTER, 1/4, the area!
Now, there are a couple of potential options to correct this discrepancy. First, if you have some kind of "original" source image that you created your first 300dpi printable image from, you can go back to that original content, create another copy, and scale it to the right physical print dimensions (width and height in inches, or if you prefer metric). If your original source image is a very high resolution image, then you can RESAMPLE (enable that option again), as it should be DOWNsampling. Downsampling from a high resolution original when printing to a smaller size is usually fine, as you have more information than you need.
Now, if you don't have any kind of original source image, you can try to UPsample the image you have. Upsampling (resampling from a smaller image to a larger) has to, in essence, fabricate information...or interpolate between sparser pixels of a lower resolution image into a higher resolution image. This is an imperfect process...so you won't, necessarily, get a more detailed image than at 300dpi. With modern upscaling algorithms, or even AI powered ones, you can potentially preserve a lot of the details...but for maximum detail, it is BEST to start with a clone of a truly high resolution image, then DOWNSAMPLE instead.
So what you would do if your dimensions are incorrect (regardless of up or down sampling), is instead of unchecking the Resample option, you would leave it enabled. You can keep it on "Preserve Details". The next most important thing, is that you set the MINIMUM physical print dimension, to what you require (and depending on exactly how you are going to deal with your seam, maybe slightly larger to allow for a tiny amount of the "bleed" (the very edge border area of your image) to be trimmed away for a real clean edge. You need to make sure the smaller dimension will at the very least print the right size.
If the RATIO of the image, the larger dimension over the smaller, is correct, then your larger dimension should also end up perfect. If not, then once you have resampled your image to 600dpi and the desired print dimensions, you will need to CROP the image such that the print dimensions are correct for both the longer and shorter edges. You can either do this in Photoshop with its cropping tool, or...you can simply trim down to the desired size in the physical print. IMO, it is easier to use the PS crop tool, as it can allow you to see how your print will be framed, before you actually trim out any excess. It will also give you some dynamic room to maneuver and figure out exactly which part of your image, you really want in your label. Further, if you have any seam-mitigation work to do, you can do that in the cropped image in PS, by working to get the seams of the longer edges to match up however you prefer.
Once you have the DPI and physical dimensions set...then you can print, and it should print out at the detail you desire.
