A pump should definitely be taken on any sort of bike ride you take, although storing it can sound a lot harder than it actual is.
Mount it on the frame
One of the simplest, although not the most aesthetically pleasing ways to carry a pump on your bike is by mounting it on the frame.
There are a load of ways to do that, but the most common is probably by securing it to the bottle cage bracket. Depending on the type, the pump doesn’t always take up the water bottle’s space, so there’s that.
What makes this approach pretty great is that you’ll be able to mount bigger pumps, which will make inflating tires much quicker, but sadly this solution doesn’t come without any disadvantages.
The pump will be prone to get covered in dirt, especially if you do mountain biking, and that can make it not function at all when it’s needed to the most.
A pump mounted this way can be easily stolen. Top of the line mini pumps aren’t that much expensive, but losing the stuff we use daily can obviously still hurt.
Carry the pump in the hydration pack
Hydration packs come in all sorts of different types and sizes, so getting one with a large storage compartment can let you hold far more than just a pump, like the multi tool, CO2 cartridges, spare tubes and such. It will even end up freeing the need to use the water bottle holder!
These types of backpacks are generally used the most in the mountain biking community, but that doesn’t mean you can’t have one if you’re a road biker, as long as it’s not an official race.
Saddle bag
Saddle bags are not the largest bicycle storage compartments out there, but definitely more accepted in the overall cycling industry.
You’ll still be able to carry a lot of various fixing equipment, but when it comes to the mini pump itself, it’ll definitely need to be more compact in order to fit the bag’s size.
Jersey pocket
If you’re a roadie, you probably already know that putting things inside your jersey’s pocket is arguably one of the best ways you can carry additional stuff without impacting your aerodynamics as much, so storing a compact pump there can be a great solution.
Storage bottle
Storage bottles can be placed inside the water bottle bracket serving as an easy to access way to carry any cycling goods you like, like a pump in this case.
Summary
Personally i think a hydration pack is the way to carry a mini pump for MTB’ers, and the jersey pocket if you’re a road biker.
Whatever you end up choosing, i highly suggest primarily using CO2 cartridges for inflating the tires when time is of the essence, and a pump, preferably the smallest one possible, just in case or when you want to fill a custom amount of air.