First, no, you should not put more emphasis on your chest in order to solve the problem. What that would entail is spot reducing (working a particular part of your body in order to reduce the fat covering that muscle) and is not really possible.
Second, if your condition is simply excess fat on your chest, then just stick with the starting strength, keep eating clean, and you will see results all over your body. Again, the fat usually comes off all over and not so much in specific places (there are exceptions obviously; some people definitely lose fat in certain areas more quickly than in others).
Third, if you have a condition called gynecomastia, then unfortunately your condition will not get any better through diet and exercise and will require surgery in order to fix. Gyno is caused by excess tissue under the nipple area, and is not a result of excess fat. If it were fat, it could be methodically burned away through diet and exercise (again, as regarding the exercise part, NOT by doing a ton of pushups or pec movements). And because it is tissue, it needs to be surgically cut out.
This condition is a pain in the ass and is actually quite common. Just look around and you will probably see what I mean.
In conclusion it is impossible to tell if you have gyno or simply too much fat. But either way, don't go messing with a proven program in order to solve this problem. If it is gyno, you can at least try to develop a decent physique all-around so overall, you look better. Good luck