[PR #9600/e6187f6 backport][3.11] Avoid starting connection timeout when a connection is already available

[bdraco]

(cherry picked from commit e6187f6)

[codspeed-hq]

CodSpeed Performance Report

Merging #9607 will not alter performance

_{Comparing patchback/backports/3.11/e6187f6808d647243d8cee115ec24c3eb1421183/pr-9600 (01c9704) with 3.11 (e86b72f)}

Summary

✅ 7 untouched benchmarks

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.87%. Comparing base (e86b72f) to head (01c9704).
Report is 254 commits behind head on 3.11.

✅ All tests successful. No failed tests found.

Additional details and impacted files

@@           Coverage Diff           @@
##             3.11    #9607   +/-   ##
=======================================
  Coverage   97.86%   97.87%           
=======================================
  Files         114      114           
  Lines       35373    35380    +7     
  Branches     4241     4243    +2     
=======================================
+ Hits        34619    34628    +9     
+ Misses        563      561    -2     
  Partials      191      191           

Flag	Coverage Δ
CI-GHA	97.75% <100.00%> (+<0.01%)	⬆️
OS-Linux	97.45% <100.00%> (+<0.01%)	⬆️
OS-Windows	94.36% <100.00%> (+<0.01%)	⬆️
OS-macOS	96.67% <100.00%> (+<0.01%)	⬆️
Py-3.10.11	96.53% <97.43%> (+<0.01%)	⬆️
Py-3.10.15	96.94% <97.43%> (+<0.01%)	⬆️
Py-3.11.10	97.02% <97.43%> (+<0.01%)	⬆️
Py-3.11.9	96.62% <97.43%> (+0.01%)	⬆️
Py-3.12.7	97.48% <100.00%> (+<0.01%)	⬆️
Py-3.13.0	97.47% <100.00%> (+<0.01%)	⬆️
Py-3.9.13	96.45% <100.00%> (+<0.01%)	⬆️
Py-3.9.20	96.86% <100.00%> (+<0.01%)	⬆️
Py-pypy7.3.16	96.46% <100.00%> (+<0.01%)	⬆️
VM-macos	96.67% <100.00%> (+<0.01%)	⬆️
VM-ubuntu	97.45% <100.00%> (+<0.01%)	⬆️
VM-windows	94.36% <100.00%> (+<0.01%)	⬆️

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[bdraco]

production retest 👍