Question: *In the context of helping someone out in driving*

1. Should it be 'ease me on out' or can I simply say 'eased me out'?
2. Finally, is my constuction correct?

For Example

> 
> I'm new to driving a car. My cousin *eased me on out* to ride through the narrow street.
> 
> 
> 

**I'm trying to say this**: When I was learning to drive the car I got stuck in the middle of the street. It was not quite easy for me to drive futher without hitting any obtacle that come in the way. So my cousin helped me out and drew the car till we were off the narrow street.

Answer: 
In the context of driving, the verb *ease* describes slow, careful movement, and the term *eased out* describes slowly merging out of some place, while *eased out on* (or *eased out into*) can mean merging onto a busy street, or into traffic. The person doing the "easing out" is generally the driver. 

It's not clear to me if your cousin was giving you advice to help you off the narrow street, or if he actually got in the driver's seat to get you out of a tricky driving situation. 

If your cousin was driving, here's a sentence you could use:

> 
> I'm new to driving a car, so **my cousin had to ease us out** so we could get through the narrow street.
> 
> 
> 

If he was just giving advice, you might say:

> 
> I'm new to driving a car, so **my cousin helped me ease out** into the narrow street.
> 
> 
> 

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Here are a few more examples; hopefully, they can help you better understand how to use the verb *ease* in relation to driving:

[![enter image description here](https://i.stack.imgur.com/yxCqU.jpg)](https://i.stack.imgur.com/yxCqU.jpg)

> 
> *The driver **eased the car** through the narrow street, being careful not to hit the pedestrians*.
> 
> 
> 

When we use *ease out*, it means the car is going "out" somewhere, like out of a parking spot, or out onto a highway: 

[![enter image description here](https://i.stack.imgur.com/BO26k.jpg)](https://i.stack.imgur.com/BO26k.jpg)

> 
> *The driver of the silver car is **easing out** into the rush hour traffic*.
> 
> 
> 

We can also *ease into* instead of *ease out on*:

[![enter image description here](https://i.stack.imgur.com/7o4cz.jpg)](https://i.stack.imgur.com/7o4cz.jpg)

> 
> *The silver Fiat **eased into** a tight space between a minivan and a pickup truck*. 
> 
> 
>
Using *on* is kind of like adding extra intent or assistance.

> 
> **Come out** to the party tonight. (A simple request)
> 
> 
> **Come on out** to the party tonight. (I would like for you to go to the party)
> 
> 
> 

but either way is grammatically correct. 

However in your example, "eased me on out to ride" doesn't make sense, because "eased me out to ride" doesn't either. You might need to explain your logic and what you are trying to say.