There are many ways to do that. First you need to find the custom field ID.
Quick and easy
Declarative and quick way to do that will be either looking at the source code of the page (ctrl/command + U) or using a developer tool extension for your browser (in this example Firebug for Firefox). Hover on the field or find the field name in the source code and you'll notice the field ID parameter as per the screenshot below:
Since this field is a lookup, in the URL we need to specify 2 values, one for the name and another for the record ID that you want to look up to. So your URL should look like
YourExistingUrlPath&CF00N9000000DMCnR=RecordName&CF00N9000000DMCnR_lkid=001900000136LSj. Note the
_lkid which means lookup ID and references the record ID of the record that we want to pre populate the field with.
Note that this is a quick way to find the field ID and you should not be hard coding this ID in your button URL because the field will have a different ID when you deploy it or create it into another org. This is only good for performing a quick test.
The dynamic programmatic way
Prior to the tooling api becoming GA, a few of us came up with the idea to dynamically parse the source code of the page in APEX and run some regular expressions in order to find the field ID. I will not give you the code for that, it's considered a bad hack :). Instead here are a few links where you can see how that is achieved now with the tooling API. If you're not a coder then you'll need a developer's assistance for this.
Querying Custom Object and Field IDs via Tooling API
How to get the entity ID for a custom field in Apex?