Wiring a Key Switch to Control USB Power?
Since you are supplying power down the USB cable to your drive, I am assuming yours is a 2.5" external hard disk drive, which only gets its power via USB (sometimes called "portable" USB hard drives).


  I'm using a USB drive, and the .7 ohms prevents readability of the drive


Yes, that result can easily happen, depending on the specific external hard drive, the current-supply capabilities of the specific PC/laptop USB port, and the specific USB cables used. Or to put it another way - some people have problems, some don't, due to different components being used.

Some USB-powered hard drives are very sensitive to any voltage drop, and can fail to operate when other (lower power) equipment works OK. At power-on, a typical 2.5" hard disk drive which is not designed for low-current power-on, exceeds the maximum current from a single USB 2.x port by a significant amount (e.g. briefly requiring >1 A during power-on vs. the 500 mA maximum from a USB 2.x port, and even that doesn't have to be available without USB enumeration, though it usually is).

These drives only power-on at all due to capacitors, luck, and USB ports supplying more current than the USB specification requires - or Y-cables!

Some other 2.5" drives can be configured for a low-current power-on, where the drive limits its current required to around 500 mA during spin-up, at the cost of having a longer spin-up phase. However those drives still have limits for acceptable voltage drop and hence limits for any significant resistance in the power cables.


  My initial solution was to solder a button cell battery to the cable to provide the extra volts.


This won't work. Such cells cannot supply the necessary current.


  My problem is that this specific switch has a .7 ohm resistance.


That is only part of your problem. In addition to the resistance of the switch:


You have also added extra resistance by using a USB extension cable. We can't see how long it is in the photo, but its power wires look thin, so it isn't one of the cables with specially thicker power wires. That additional cable will increase the voltage drop measured at the drive itself. 
You have removed the shielding from (what looks like) 4"-6" of the USB data pair in the cable. That could increase data-related problems caused by external interference. It is good that you didn't untwist the wires in the USB data pair; that would have been even worse.
It also appears that you have broken the end-to-end continuity of the cable's shielding. Again, that will not help reliable USB data communications.
Switching-off power to USB-powered hard drives should not be done lightly. If the disk is spinning when the power is switched-off, it forces an "emergency retract" which is a high-stress mechanical event for the drive and can cause data corruption to any writes-in-progress, and loss of any pending writes.


If you insist on having a power switch like this, then use one of the previous suggestions e.g. a switch-operated carefully-chosen relay or MOSFET (selected to minimise the added resistance), a very short USB extension cable preferably with thicker power wires e.g. 20AWG, removing the minimum of USB shielding, and perhaps it might work. But since success depends on the specific requirements of that USB-powered hard drive and the PC's USB port, there is no guarantee that the overall configuration can cope with any additional resistance at all.

Of course for normal low USB power situations, a Y-cable might be suggested. However if you are trying to switch the power, then these cables add to the complexity, as you then need to switch the power coming from two USB ports instead of one!