diff --git a/agent/chat_completion_helpers.py b/agent/chat_completion_helpers.py
index 45a88f8decf0..ad2c3cfae229 100644
--- a/agent/chat_completion_helpers.py
+++ b/agent/chat_completion_helpers.py
@@ -17,6 +17,7 @@ from __future__ import annotations
 
 import json
 import logging
+import math
 import os
 import re
 import threading
@@ -191,6 +192,30 @@ def _env_float(name: str, default: float) -> float:
         return default
 
 
+def _codex_wait_notice_recovery(
+    *,
+    stale_timeout: float,
+    ttfb_enabled: bool,
+    ttfb_timeout: float,
+    last_event_ts: Optional[float],
+    call_start: float,
+    idle_enabled: bool,
+    idle_timeout: float,
+) -> str:
+    """Describe the earliest enabled Codex watchdog on the call timeline."""
+    deadlines: list[float] = []
+    if math.isfinite(stale_timeout):
+        deadlines.append(stale_timeout)
+    if last_event_ts is None:
+        if ttfb_enabled and math.isfinite(ttfb_timeout):
+            deadlines.append(ttfb_timeout)
+    elif idle_enabled and math.isfinite(idle_timeout):
+        deadlines.append(max(0.0, last_event_ts - call_start) + idle_timeout)
+    if not deadlines:
+        return ""
+    return f"; auto-reconnect at {int(min(deadlines))}s"
+
+
 # ── Cross-turn stale-call circuit breaker (#58962) ─────────────────────
 # A session wedged against an unresponsive provider hits the stale detector
 # on every call and loops forever (observed: 494 consecutive failures over
@@ -611,16 +636,19 @@ def interruptible_api_call(agent, api_kwargs: dict):
         # usually a slow/overloaded provider, but the UI never said so).
         if _poll_count % 100 == 0:  # 100 × 0.3s = 30s
             _elapsed = time.time() - _call_start
-            _deadline = _stale_timeout
-            if (
-                _ttfb_enabled
-                and getattr(agent, "_codex_stream_last_event_ts", None) is None
-            ):
-                _deadline = min(_deadline, _ttfb_timeout)
+            _recovery = _codex_wait_notice_recovery(
+                stale_timeout=_stale_timeout,
+                ttfb_enabled=_ttfb_enabled,
+                ttfb_timeout=_ttfb_timeout,
+                last_event_ts=getattr(agent, "_codex_stream_last_event_ts", None),
+                call_start=_call_start,
+                idle_enabled=_codex_idle_enabled,
+                idle_timeout=_codex_idle_timeout,
+            )
             agent._emit_wait_notice(
                 f"⏳ waiting on {api_kwargs.get('model', 'the provider')} — "
                 f"{int(_elapsed)}s with no response yet (provider may be slow "
-                f"or overloaded; auto-reconnect at {int(_deadline)}s)"
+                f"or overloaded{_recovery})"
             )
 
         _elapsed = time.time() - _call_start
diff --git a/tests/agent/test_codex_ttfb_watchdog.py b/tests/agent/test_codex_ttfb_watchdog.py
index 6a647164f93d..977e56335047 100644
--- a/tests/agent/test_codex_ttfb_watchdog.py
+++ b/tests/agent/test_codex_ttfb_watchdog.py
@@ -316,6 +316,24 @@ def test_event_idle_kills_after_first_event_then_silence(tmp_path, monkeypatch):
         stop["flag"] = True
 
 
+def test_wait_notice_handles_infinite_local_stale_timeout():
+    """After the first SSE event, a local endpoint's infinite wall-clock
+    timeout must not reach ``int()``; report the finite idle watchdog instead."""
+    from agent import chat_completion_helpers as h
+
+    recovery = h._codex_wait_notice_recovery(
+        stale_timeout=float("inf"),
+        ttfb_enabled=True,
+        ttfb_timeout=120.0,
+        last_event_ts=130.0,
+        call_start=100.0,
+        idle_enabled=True,
+        idle_timeout=60.0,
+    )
+
+    assert recovery == "; auto-reconnect at 90s"
+
+
 def test_ttfb_disabled_via_env_zero(tmp_path, monkeypatch):
     """Setting HERMES_CODEX_TTFB_TIMEOUT_SECONDS=0 disables the TTFB watchdog;
     a no-event stall then falls through to the (here, 60s) stale timeout, so a