The Problem
I move coins out of and into exchanges somewhat regularly. In the beginning of my cryptocurrency adventures, I would typically use Bitcoin to transfer. I mean, it was simple that way: Transfer Bitcoin from Coinbase to any other exchange I wanted and then buy the altcoins I was most interested in. As time went on, however, I began noticing I couldn't buy near as many altcoins as my math had led me to believe I could buy(which was due to increasing transaction fees). Furthermore, little by little, the transactions began taking longer and longer (which was due to increasing transaction times).
The Solution
To cut to the chase, I alway use Litecoin to transfer because there are virtually no tx fees (in fact, LTC's tx fees are some of the lowest among all cryptocurrencies), and it’s WAY faster than BTC. So, if you’re holding Bitcoin, you can exchange it for LTC for the fee (this is actually free if you use a Limit Order within GDAX), which is usually only about 0.15% to 0.25% on most exchanges (so, far lower than Bitcoin’s tx fee), and then transfer it to the exchange of your choice (and because you’re using LTC, the tx fee, again, is incredibly small to transfer coins out of an exchange). Once the coin arrives at the other exchange, sell LTC for Bitcoin, again incurring a small loss. Then, buy your altcoins with BTC.
As you can see, it is a slightly longer process, but in the end, you've saved a ton with smaller transaction fees. And, I don't know about you, but every minute that goes by during a transfer from one exchange to another is STRESSFUL. I always think I sent it to the wrong address. So, the fact that LTC transfers are far faster is so much less stressful and burdensome.
One Caveat
Now, this really only works if your intention is to purchase another altcoin, though, because of capital gains. So, if you were already planning on selling BTC for another coin, you’d already have to worry about them, so the exchanging between LTC and BTC wouldn’t mean much. This is just my process, though. I’m sure there are better ones out there.
The other solution is to not claim capital gains (because, hey, I'm no judge).