Note that SIT/PASI at Sitka Alaska currently (April 2022) features a large E4 surface-level Class E "extension" extending 28 miles northwest of the airport, adjoining a core E2 surface-level Class E airspace area that surrounds the airport out to a radius of 4.2 nm and also has a small projection to the southwest.1 Why the airspace was designated this way, rather than simply all designated as part of a single E2 surface-level Class E airspace area, is unknown to this answerer and is the subject of a related ASE question. At any rate, the surface-level controlled airspace northwest of SIT/PASI beyond 4.1 nm from the airport is an E4 surface-level Class E "extension", exactly as is the airspace at KSTS referenced in the present question.
I've spoken to a staffer at the FSS at Sitka Alaska, which is responsible for actually issuing a SVFR clearance for the airspace near SIT/PASI, and was informed that 1) a Special VFR clearance would be valid throughout the entire surface-level Class E airspace, including the E4 "extension", and 2) the E4 "extension" is having a negative impact on Coast Guard helicopter training operations because when the ceiling is lower than 1000' AGL in that area they can't operate VFR there without a Special VFR clearance, and since SVFR clearances are issued on a one-in one-out basis, that's not compatible with extended training operations.
So at this particular airport, the folks involved in actually issuing a SVFR clearance do consider it to be valid for the entire surface-level Class E airspace including the E4 "extension" extending 28 miles to the northwest, and also do consider FAR 91.155(c), the prohibition on VFR operations below a ceiling lower than 1000' AGL without a Special VFR clearance, to cover the entire surface-level Class E airspace, including the E4 "extension" extending 28 miles to the northwest.
Incidentally, the FSS staffer I spoke to used the term "surface area" to refer to the entire surface-level Class E airspace in this area, including the E4 "extension" extending 28 miles to the northwest.

Perusing the related links below will suggest that "your mileage may vary". The truth is that the FAA is deeply divided as to whether phrases in the FARs like "surface area", "designated for an airport", and "surface area of controlled airspace designated for an airport" should or should not be construed to include surface-level Class E "extensions".2 Ambiguity around this issue has existed ever since the 1993 "alphabet" airspace re-designation.3 Various FAA issuances in the past have given conflicting answers to this question.4 Pilots have failed check rides due to the confusion and ambiguity around this issue within the Agency.5 There's recently been some consideration of a future rule-making effort to address this issue.6 It's possible that the language in AIM 3-2-6, as well as in the legend of the Chart Supplements, that reads "Surface area arrival extensions become part of the surface area and are in effect during the same times as the surface area" will be changed in the future to avoid implying that all regulations that apply to the core "surface area" should automatically be construed to apply to the surface-level Class E "extensions" as well.7
Some of the answers to the ASE questions linked below--particularly the last three questions in the list -- delve into the specifics of the phraseology in various FARs and in the "Airspace Designations and Reporting Points" document, and explore why it is arguably unclear as to whether phrases like "within the lateral boundaries of controlled airspace designated to the surface for an airport" (the specific phrase used in FAR 91.155(c), "within the lateral boundaries of the surface area of Class E airspace designated for an airport", and other similar phrases should or should not be construed to include surface-level Class E "extensions" (E3/E4 airspaces).
Clearly, the most conservative and safest course of action for pilots is to assume that FAR 91.155(c) should be considered to apply to E4 surface-level Class E "extensions", not just to the core "surface areas" that actually surround the airports for which they are designated--regardless of whether or not the controlling agency in any given instance is actually willing to issue you a SVFR clearance for flight within the E4 "extensions". Also, whenever a pilot is issued a SVFR clearance for a given controlled airspace, he or she should verify whether or not the controlling agency intends that SVFR clearance to include any adjoining "extensions", in any case where that might be a factor.8
Footnotes:
- The E4 surface-level Class E "extension" at SIT/PASI (Sitka, Alaska) is 12 miles wide, and extends to 28.3 miles northwest of the airport. For the E-4 airspace description, see page E-158 of the "Airspace Designations and Reporting document FAA Order 7400.11F. The E-2 airspace description appears on page E-9. Both of these airspace descriptions were last modified on 11-05-2020, which is when the large E4 "extension" was introduced.
2,3,5,6,7,8. Source-- comments made by high-level FAA staffers from Airspace and Flight Standards divisions, during April 2021 airspace charting group meeting.
- See related ASE answer What indication has the FAA given that phrases like "surface area of Class E airspace designated for an airport" do or don't include E4 "extensions"?
Related ASE questions:
Q: Why does the surface-level Class E airspace at SIT/PASI (Sitka Alaska) extend so far to the northwest?
Q: In the US, in actual practice, workload permitting, will ARTC facilities grant SVFR clearance for surface-level Class E "extensions" (E4 airspace)?
Q: Does an SVFR clearance extend to Echo surface extensions?
Q: What indication has the FAA given that phrases like "surface area of Class E airspace designated for an airport" do or don't include E4 "extensions"?
Q: "Do the rule-making docs preceding the 9-16-93 "alphabet" airspace re-designation indicate that these phrases were meant to exclude E3/E4 'extensions'?" (Link)