diff --git a/src/hip/_connection.py b/src/hip/_connection.py
index f8f4f4c..5e36a47 100644
--- a/src/hip/_connection.py
+++ b/src/hip/_connection.py
@@ -19,7 +19,6 @@
     "Connection",
     "ResponseStream",
     "Transport",
-    "open_connection",
 ]
 
 
@@ -80,11 +79,30 @@ def stream(
 
 
 class Connection(Transport):
-    def __init__(self, stream: Stream, origin: URL | str):
-        self._stream = stream
+    def __init__(self, origin: URL | str, stream: Stream | None = None):
+        self._stream = Stream() if (stream is None) else stream
         self._origin = URL(origin) if not isinstance(origin, URL) else origin
         self._request_lock = Lock()
-        self._parser = HTTPParser(stream, mode="CLIENT")
+        self._parser = HTTPParser(self._stream, mode="CLIENT")
+
+        # Ensure valid URL
+        if self._origin.scheme not in ("http", "https"):
+            raise ValueError(
+                f"Invalid URL. Expected http:// or https://, got {self._origin.scheme}://"
+            )
+
+    def open(self, backend: NetworkBackend | None = None) -> None:
+        backend = NetworkBackend() if (backend is None) else backend
+
+        scheme = self._origin.scheme
+        host = self._origin.host
+        port = self._origin.port or {"http": 80, "https": 443}[scheme]
+
+        if scheme == "https":
+            self._stream = backend.connect_tls(host, port)
+        else:
+            self._stream = backend.connect(host, port)
+        self._parser = HTTPParser(self._stream, mode="CLIENT")
 
     # API for connection pool management...
     def origin(self) -> URL:
@@ -180,6 +198,7 @@ def __repr__(self) -> str:
         return f"<Connection [{self._origin} {self.description()}]>"
 
     def __enter__(self) -> "Self":
+        self.open()
         return self
 
     def __exit__(
@@ -189,28 +208,3 @@ def __exit__(
         traceback: types.TracebackType | None = None,
     ):
         self.close()
-
-
-def open_connection(
-    url: URL | str,
-    hostname: str = "",
-    backend: NetworkBackend | None = None,
-) -> Connection:
-
-    if isinstance(url, str):
-        url = URL(url)
-
-    if url.scheme not in ("http", "https"):
-        raise ValueError("URL scheme must be 'http://' or 'https://'.")
-    if backend is None:
-        backend = NetworkBackend()
-
-    host = url.host
-    port = url.port or {"http": 80, "https": 443}[url.scheme]
-
-    if url.scheme == "https":
-        stream = backend.connect_tls(host, port, hostname)
-    else:
-        stream = backend.connect(host, port)
-
-    return Connection(stream, url)
diff --git a/src/hip/_pool.py b/src/hip/_pool.py
index 812056c..0527a2c 100644
--- a/src/hip/_pool.py
+++ b/src/hip/_pool.py
@@ -6,7 +6,7 @@
     # Once we drop 3.10 support, we can use 'typing.Self'.
     from typing_extensions import Self
 
-from ._connection import Connection, ResponseStream, Transport, open_connection
+from ._connection import Connection, ResponseStream, Transport
 from ._request import Request
 from ._urls import URL
 from .backends.sync import NetworkBackend, Semaphore
@@ -53,9 +53,8 @@ def _get_connection(self, request: Request) -> "Connection":
                 return conn
 
         # Or else create a new connection.
-        conn = open_connection(
-            origin, hostname=request.headers["Host"], backend=self._network_backend
-        )
+        conn = Connection(origin)
+        conn.open(backend=self._network_backend)
         self._connections.append(conn)
         return conn
 
diff --git a/src/hip/transports.py b/src/hip/transports.py
index e1e24a4..4ec7c0d 100644
--- a/src/hip/transports.py
+++ b/src/hip/transports.py
@@ -1,4 +1,4 @@
-from ._connection import Connection, ResponseStream, Transport, open_connection
+from ._connection import Connection, ResponseStream, Transport
 from ._pool import ConnectionPool
 
 __all__ = [
@@ -6,5 +6,4 @@
     "ConnectionPool",
     "ResponseStream",
     "Transport",
-    "open_connection",
 ]
diff --git a/tests/test_pool.py b/tests/test_pool.py
index b48f71e..b3978a1 100644
--- a/tests/test_pool.py
+++ b/tests/test_pool.py
@@ -80,7 +80,7 @@ def test_connection_pool_should_have_managed_lifespan(server):
 
 
 def test_connection_request(server):
-    with hip.transports.open_connection(server.url) as conn:
+    with hip.transports.Connection(server.url) as conn:
         assert repr(conn) == f"<Connection [{server.url} idle]>"
 
         r = conn.request("GET", "/")
@@ -90,7 +90,7 @@ def test_connection_request(server):
 
 
 def test_connection_stream(server):
-    with hip.transports.open_connection(server.url) as conn:
+    with hip.transports.Connection(server.url) as conn:
         assert repr(conn) == f"<Connection [{server.url} idle]>"
         with conn.stream("GET", "/") as r:
             assert r.status_code == 200
@@ -101,7 +101,7 @@ def test_connection_stream(server):
 
 def test_connection_stream_read_in_chunks(server):
     with (
-        hip.transports.open_connection(server.url) as conn,
+        hip.transports.Connection(server.url) as conn,
         conn.stream("GET", "/") as response,
     ):
         assert response.read(5) == b"Hello"
@@ -110,19 +110,17 @@ def test_connection_stream_read_in_chunks(server):
 
 
 class TLSBackend:
-    def connect_tls(self, host, port, hostname):
-        self.connect_tls_args = (host, port, hostname)
+    def connect_tls(self, host, port):
+        self.connect_tls_args = (host, port)
         return hip.Binary(b"").open()
 
 
 def test_open_https_connection():
     backend = TLSBackend()
 
-    with hip.transports.open_connection(
-        "https://example.com",
-        hostname="example.com",
-        backend=backend,
-    ):
-        pass
-
-    assert backend.connect_tls_args == ("example.com", 443, "example.com")
+    conn = hip.transports.Connection("https://example.com")
+    conn.open(backend=backend)
+    try:
+        assert backend.connect_tls_args == ("example.com", 443)
+    finally:
+        conn.close()