While our top solution and best solution may be the most efficient, maybe the most technologically sound, but in reality, you have to match those attributes to the attributes that is involved in solving your problem. These are the attributes of the solution. As much as we want to find the solution and solve the problem, we also have be aware of what makes a good solution. These are the solution's attributes.
Situation where this arises, is when the solution that we think is number one, the best and the right one, may not be agreeable to everybody or the stakeholders who have a vested interest in their solution. We should also take a look, as closely as we do the first one, the second solution. Note what are the attributes that make it a good solution. And what are the attributes that don't make it a good solution? Those that are less better than the number one option.
Quite often problem solvers are enamored by the most complex or elegant solution. They forget that there are other factors to take into consideration. Like cost. Like time.
Your number two solution may not be the most elegant, nor the fastest, but it could be the cheapest. It may take longer, but if it doesn't go over the threshold time limit, do a comparison of the final cost of generating the solution or executing the solution. Sometimes, a problem which doesn't 'have a solution' will suddenly find one the in the second best solution.