Installing another pump after the primary pump may damage the primary pump, but even if it doesn't it won't solve your problem. The bigger issue is that if the line between the two pumps has any leaks, even microscopic, then you'll be creating a very low pressure in that pipe, which may lead to contamination. You would have to make sure the secondary pump is lower pressure, and this would defeat any possible benefit anyway.
Consider installing a low flow shower head. These are designed for a lower volume of water, and if you can reduce the volume requirements, the apparent pressure will go up. It will provide for a more powerful shower, without having to increase the actual volume or pressure of the existing system.
Beyond that, the only way to get more volume through an existing pipe is to increase the pressure. I don't suggest that you pursue this, though, as you may create or enlarge leaks in the piping you don't want to replace.
As others have indicated, the next step is either replacing the pipe, or adding storage closer to the use.
Keep in mind that even an old high flow shower requires less than 10 gallons of water per ten minutes of showering, and during that time your source would be able to fill the storage, though not as fast as you're using it. So even a tiny 5 gallon pressure tank might give you a full pressure 10 minute shower, using 10 gallons before it goes back to the original pump volume/pressure limitations.
Add in a low flow shower head, and you could use a 5 gallon tank for 40-60 minutes before experiencing reduced pressure.
Further, a pressure tank wouldn't require a secondary pump, and would be cheaper than a secondary pump.
Placing it in the attic isn't difficult - these 5-10 gallon tanks are small, and even when full of water less than the weight of an adult. Adding a box around one with adequate insulation, and removing the insulation below it so it's more exposed to the heat from the living area shouldn't incur excessive costs.
It may be tempting to get a larger tank, but for this one problem it's probably not necessary.