The race condition occurs because the state transitions (checking if a job exists, reading its data, writing the result, and deleting the job) are not treated as a single, atomic operation. Multiple threads could simultaneously access and modify `self.jobs` and `self.results`, leading to lost updates, corrupted data, or unexpected exceptions (like trying to access a key that was just deleted by another thread).

We must use a `threading.Lock` to ensure that only one thread can execute the critical sections of code at any given time.

Here is the complete, fixed implementation:

```python
import threading

class JobQueue:
    def __init__(self):
        self.jobs = {}
        self.results = {}
        # Initialize the lock to protect shared state (self.jobs and self.results)
        self._lock = threading.Lock()

    def add_job(self, job_id, data):
        """Adds a job to the queue."""
        with self._lock:
            self.jobs[job_id] = data
        return job_id

    def process_job(self, job_id, processor):
        """
        Retrieves a job, processes it, and stores the result atomically.
        """
        data = None
        
        # Step 1: Atomically check for and retrieve the job data
        with self._lock:
            if job_id not in self.jobs:
                return False
            # Copy the data so the processor runs outside the lock 
            # (to avoid blocking the queue for the duration of the computation)
            data = self.jobs[job_id]
            # Immediately remove the job from the queue while holding the lock
            del self.jobs[job_id]

        # Step 2: Process the job (this happens outside the lock)
        if data is None:
             # Should not happen if logic is correct, but good practice
             return False
             
        result = processor(data)

        # Step 3: Atomically store the result
        with self._lock:
            self.results[job_id] = result
            
        return True

    def get_result(self, job_id):
        """Retrieves a result safely."""
        with self._lock:
            return self.results.get(job_id)

# Example Usage (Demonstration of safety)
if __name__ == '__main__':
    queue = JobQueue()
    
    # Simple processor function
    def simple_processor(data):
        import time
        time.sleep(0.01) # Simulate work
        return f"Processed: {data}"

    job_id_1 = queue.add_job(1, "Task A")
    job_id_2 = queue.add_job(2, "Task B")

    def worker(job_id, queue_instance, processor):
        if queue_instance.process_job(job_id, processor):
            print(f"Worker finished job {job_id}")
        else:
            print(f"Worker failed to find job {job_id}")

    t1 = threading.Thread(target=worker, args=(job_id_1, queue, simple_processor))
    t2 = threading.Thread(target=worker, args=(job_id_2, queue, simple_processor))

    t1.start()
    t2.start()

    t1.join()
    t2.join()

    print("\n--- Results ---")
    print(f"Result 1: {queue.get_result(job_id_1)}")
    print(f"Result 2: {queue.get_result(job_id_2)}")
    print(f"Jobs remaining in queue: {len(queue.jobs)}") # Should be 0
```

### Key Fixes and Explanation:

1.  **`self._lock = threading.Lock()`:** A lock object is initialized in `__init__`.
2.  **Using `with self._lock:`:** This is the idiomatic and safest way to use locks in Python. When execution enters the `with` block, the lock is acquired. When execution leaves the block (even if an exception occurs), the lock is automatically released.
3.  **Atomic Operations in `process_job`:**
    *   The most critical fix was restructuring `process_job`. We now use the lock to perform the **check-and-delete** operation atomically (Step 1). This ensures that no other thread can interfere between the check (`if job_id not in self.jobs`) and the deletion (`del self.jobs[job_id]`).
    *   The time-consuming processing (`processor(data)`) is done **outside** the lock. This is crucial for performance; if we kept the lock held during the processor execution, the entire queue would serialize, defeating the purpose of concurrent processing.
    *   The final result storage (Step 3) is protected by a separate lock acquisition to ensure the write to `self.results` is safe.

By protecting all accesses and modifications to `self.jobs` and `self.results` with locks, we guarantee that the shared state remains consistent, resolving the race condition.