Because an options spread has more than one "leg," does it make sense to enter one leg at a time if you can get a better price on a different exchange? Maybe.
In the world of trading, the term “legging” refers to putting on the different parts of an option spread as individual transactions. For example, a put vertical spread is composed of a long put at one strike and a short put at another. The long is one “leg” of the spread, and the short is the other. You could establish a position in the put vertical as a single transaction. Or you could put an order in to buy the long put, wait for a fill, then enter an order to sell the short put.
The main reason for legging is to try to get a better fill price. But this can lead to problems.
Say you want to sell a call vertical for a net 1.00 credit, with the short call at 1.50 and the long call at .50, because you think the stock might go down longer term. But intraday, you think the stock might bounce higher. So you try to leg into the short call spread by buying the long call leg first for $0.50, then wait until the price goes up a bit before entering the order to sell the short call leg.
If the stock does go up, you might be able to sell that call at a higher price, say 1.75, and get 1.25 credit for your short call vertical. In this case, you'd be getting a higher net credit by legging into the call spread than you could have selling the call spread as a single transaction.
If the stock drops instead of rises, then the call you're looking to sell to establish the short call vertical could be dropping in price, too. You'd be getting a worse price for your short call vertical than if you sold the spread as a single transaction.
If you're very confident in your speculations, have the discipline to trade that second leg if the stock goes against you and you have assessed the risk and possible loss, then you may consider legging with a few caveats.
Do the long leg first. Imagine you tried legging into a short call vertical with the short call leg first, and the stock suddenly jumped higher. You'd have a growing loss on that short call option, and possibly a margin call, and you'd have to try to buy the long call leg at higher prices. If your first leg is a short option, the margining systems don't know you intend to buy an option against that short, so it looks at that short option order as naked. That means the order could be rejected, or significantly reduce the buying power in your account.
Don't leg into spreads with more than two legs. Trying to keep track of the prices and not getting legged out on complex spread is very hard, even for pros. Do the “hard” leg first. First try to get filled on the option that might have a wider bid/ask spread, or lower liquidity (as measured by volume and open interest). If you do the “easy” side first, with the option that is more actively traded, you could get filled very quickly, but then have a hard time getting the other, less actively traded leg filled.
Don't be stubborn. If you get legged out, you must exercise discipline to get the other side of the spread on, even if the price has moved against you. Don't be greedy or overconfident—the option might not come back to where you want to trade it. Have a price for either the stock or the option you'll use to decide to either put the second leg on, even if it's a worse price, or close out the first leg for a loss. That can limit potential losses while legging.