I'm looking for a frame pump with a bracket that can be mounted in between the bottle cage and the frame, so that the cage can stay mounted, and the frame pump would be situated right beside the cage. Does anything like that exist? I've seen some brackets that replace a cage, but not any that can be used together with the cage using the same mounting bolts.

That's exactly how mine is mounted. The pump mount is slightly offset. The bottle cage and pump are mounted under one set of screws.
It's this one.

There's quite a few.
I have this one:

https://www.blackburndesign.com/pumps...l#.UmV0RihwaQI

Wait, what are you looking for? A "frame pump," or more commonly perhaps a "frame-fit" is sized precisely to fit the frame, and as such does not use a bracket.

FenderTL5 posted what is properly called a "mini pump," and just about every mini pump comes with a bracket designed to mount under the cage, so I'm confused as to what you're looking for.

Used to be a bracket that let you mount the bottle cage ahead of a seat-tube mounted frame-fit pump
.. a proper one that is as long as possible for your frame size.

Pumping up tires goes faster the more air you can move , per stroke.

I have a Lezyne that mounts that way. However, I keep it in my bag so it doesn't get stolen if I forget to remove.

You can get the brackets separately from the pumps so if you're interested in a pump that doesn't come with a bracket (most seem to include it though), you can get a bracket.

Yeah I'd like something about the size of a Road Morph G, just not sure if that one comes with a bracket that will let me retain both cages. I don't really want to strap the pump directly to the frame, since both my top and down tubes have cables running alongside, and I like to avoid strapping stuff around the cables.

I think you mean a mini-pump, not a frame pump.

Anyway, yes. I have one like this, and I bought an extra bracket so I can move it between two bikes. Here it is:

https://www.amazon.com/Lezyne-Pressur...pressure+drive

Specialized has a pump that's similar to the Lezyne...it's called the Air Tool, available in multiple PSI capacities. I mention it because Spesh stuff might be easier to find in some areas.

Patrick check out the Lezyne line of pumps. They mount just as you want it and they are small and lightweight. An added bonus is that the end of the hose threads onto the valve, thus avoiding the dreaded "torn valve syndrome."

Check them out.

That Road Morph G is a pretty chunky pump; maybe you should look at the Lezyne Road Drive Large, which gives up only about 2" in overall length, but is slender enough to use the included the side-mount braze-on bracket mounted behind a cage.

I'm sure the RMG may be more ergonomically comfy, what with the pivoting handle and foot peg, but you have to sacrifice a lot of space to get it. A real frame-fit, e.g. Topeak Master Blaster or Park PMP-5, wouldn't require straps to mount, while providing "full size pumping action" (Sounds dirty, doesn't it?!).

As an extra added bonus, the optional Lezyne Pen Gauge replacement hose lets you get the pressure just right!

Yes, I forgot to mention that. The gauge is fairly accurate too.

Another vote for Lyzyne - I have the Lezyne Road Drive Hand Pump, its bracket fits well under my bottle cage and it an pump up road tires to a high PSI if that is important for you.

https://www.amazon.com/Lezyne-Road-Dr...2386749&sr=1-3

I also noticed they sell a special integrated bottle cage/pump holder, but not sure thats needed from my experience. The pen gauge seems very interesting addition I might have to pick up.

https://www.amazon.com/Lezyne-PEN-GAU...ef=pd_sim_sg_1

The Road Morph G comes with a bracket that replaces a bottle cage. However, Topeak also sells a bracket for that pump that fits under the cage. I have the Road Morph on my commuters but use the Lezyne (w/pen gage sold separately) medium pressure drive pump with my road bike. It's mounting bracket fits under the bottle cage and came with the pump. Like them both.

I have a couple of Lezyne's, a Blackburn and a Road Morph G mini. Most of the bikes have Lezyne brackets under the bottle cage and one has the Blackburn bracket. They all seem to fit in all the brackets, though I don't trust anything but the Blackburn in the Blackburn bracket because there is no velcro to batten it down. I don't like the Road Morph because it is so big it gets in my way.

Oooo. I need one of those!