I have participated in a somewhat different approach to evangelism, and it kinda goes along the lines of what Jesus did, as when he preached, the kingdom of God manifested itself then and there.
We set up signs that said "Free Spiritual Readings" at a Booster Juice place, some people gave out flyers or just told people that if they wanted a free spiritual reading that they could go there. It was summer and I live in a place with lots of tourists.
People lined up half way down the block to get in and we would ask God for what he wanted to say to each person, and he would give very specific words of knowledge and other things that he wanted to say to them.
We introduced ourselves as Christians and asked them if they wanted to know anything specific or if they simply wanted us to ask God what he would have to say to them. In my case most chose the second option.
At the end of the day, there were many people who were deeply ministered to and found healing from past hurts and found new hope in Jesus. Examples of what was happening: a body builder crying at one of the tables, my partner had a word of knowledge about how the girl in front of us was sexually abused at 13 and told her all of this information that she had told no one about and at that point she knew that she had come into contact with God, and was an open book and received some healing from Jesus that afternoon.
Most of the people that I had were Christians, to my shock, as until you were sitting at the tables and the introductions were made, there was no mention of Jesus or Christianity. People are spiritually hungry and will go anywhere to satisfy that hunger. Nevertheless, God was a strong encouragement to them as well.
SUMMARY
In my experience people do not need to have an intellectual argument for the gospel. At least where I live. Everyone has heard the message and seen the hypocrisy in some believers, and written Jesus off. By stepping out of the way and allowing the person to meet Jesus personally (the revelatory gifts are very useful here - God had told another friend of mine the names of the people in front of him! Needless to say, God had their attention at that point!) the real message and reality of it gets communicated.
