Create the session around the new players. Rather than simply having the party just bump into the new player in the street and hire them (unless the character is literally a mercenary who offers their services to any random party he thinks is worthy), have them all put into the area and working on the same mission. They should gladly work together as both sides have separate motivation to complete the quest. After the quest, the new players decide to come along for one reason or another.

They could be related to the quest giver or an important NPC in the adventure. They could be someone brought in to help as well, or who is currently stuck in the quest. Could even have the quest centered specifically around the character and something that eventually pushes them into the welcoming arms of the party. It could be a side quest, it could be a main quest. Doesn't matter.

For Example, I have prepared one quest, and three different characters and a hook into the adventure :

Quest : Mysterious Deaths

Player : Drow Rogue

The adventure leads to a small village to investigate the recent victim in a serious of mysterious deaths in the area. The source is definitely a demon on the prowl traveling around sowing chaos and discord but the party does not know this yet. They reach the village to find that the mayor is not interested in the party's help, as they caught the culprit. They COULD investigate the victim, and determine for sure that it couldn't have been the Drow. The lead the party to the prison where the Rogue is sitting there (or not, if they decided to break out.) The party encounters the new party member, and they agree to help the Drow out to clear his/her name. They find the demon, slay it, and the Drow decides to join the party as the Drow now believes that they now owe a life debt to the party.

Player : Half Paladin

The adventure leads to a small village to investigate the recent victim in a serious of mysterious deaths in the area. The source is definitely a demon on the prowl traveling around sowing chaos and discord but the party does not know this yet. They reach the village to find that the mayor is desperate for aid, she beseeched the aid of a traveling Paladin to help her solve this mystery. The Paladin has been hunting a errant demon in the area, and believes that these deaths could be related. He certainly use the extra pair of hands, and they could always use a Paladin. They find the demon, slay it, the party likes the Paladin and offers to join. The Paladin gracefully accepts.

Player : Half Elf Ranger

The adventure leads to a small village to investigate the recent victim in a serious of mysterious deaths in the area. The source is definitely a demon on the prowl traveling around sowing chaos and discord but the party does not know this yet. They reach the village to find that the mayor is desperate for aid. The spouse of the victim, a fresh Ranger is seeking revenge. The party offers the ranger the opportunity for revenge. They find the demon, slay it, the Ranger has their revenge and is at a lost to do with the rest of their life. The party offers an opportunity to find a new purpose in life within the party.