Choosing a stopover city when the cheapest routing offers a long layover

The cheapest long-haul ticket has a ten-hour daytime layover in a city I've never visited. That could be a bonus mini-trip, or a miserable stretch of security, luggage and transport stress. The airline offers no organized stopover package. How do you judge whether leaving the airport is realistic—visa rules, baggage handling, distance downtown and buffer time—before convincing yourself you've found a free holiday?