Madigan Park is a tiny community of older and newer single-family homes located on S. Hubert Ave. just off W. Kennedy Blvd. covering half the block. The other half is multi-family apartments and businesses. The housing section is beautifully covered in old oak trees for shade and very well-kept. There are pet stores within both walking and driving distance, but the nearest dog-friendly parks are further out and will require you to pack your pup in the car.
Being this close to Old Tampa Bay, the weather here is quite mild year round with temperatures hovering in the low 70s during the winter months and summer highs hanging out in the low 90s. There are very few days where you can't take your dog for a walk but beware of the temperatures of any paved surfaces during the summer months as they can reach very high temperatures which are more than hot enough to burn your pup's paws.
As you might expect being this close to the water, the land here is very flat and unforgiving. With very little in the way of greenways for your pup to walk on in the nearby area, you and your pup are going to have to get used to walking on paved surfaces. There are two issues to consider when taking your dog walking down this terrain. First, you need to keep a close eye on your dog's paws. Until they have had time to toughen them up, the paved surfaces could scrape them raw if you make your walks too long. Secondly, fall, winter, and spring are the best times to go walking as the ambient temperatures are low enough to keep the pavement from getting hot. You may want to consider investing in some doggy boots to keep your pup's paws nice and safe in the hotter months.
This is an area of Tampa in which someone forgot to add in very many green spaces for you to enjoy. The only way you are going to locate greenways large enough for you and your pup to play on is to take a drive.
The nearest park to you is Cypress Point Park where you can watch the planes coming and going from Tampa International Airport and hike along the trail that runs the length of the park all before taking your pup out for a walk out in the grass. The park also offers grills, lots of parking, and a spectacular view of the sunset over Old Tampa Bay. The only thing you can't do is take your dog to the beach as it is against the park rules. Your dog must always be kept on their leash and you must clean up after they make a mess or you face a fine.