How can I define a route differently if parameter is not integer

Just add ->where('id', '[0-9]+') to route where you want to accept number-only parameter:

Route::get('contacts/{id}', 'ContactController@get_contact')->where('id', '[0-9]+');
Route::get('contacts/new', 'ContactController@new_contact');

Read more: http://laravel.com/docs/master/routing#route-parameters

A simple solution would be to use an explicit approach.

Route::get('contacts/{id:[0-9]+}', 'ContactController@get_contact');
Route::get('contacts/new', 'ContactController@new_contact');

Although the accepted answer is perfectly fine, usually a parameter is used more than once and thus you might want to use a DRY approach by defining a pattern in your boot function in the RouteServiceProvider.php file located under app/Providers (Laravel 5.3 and onwards):

 /**
 * Define your route model bindings, pattern filters, etc.
 *
 * @return void
 */
public function boot()
{
    Route::pattern('id', '[0-9]+');

    parent::boot();
}

This way, whereever you use your {id} parameter the constraints apply.