diff --git a/Library/Homebrew/download_strategy.rb b/Library/Homebrew/download_strategy.rb
index 699228d9f0..a49b47e709 100644
--- a/Library/Homebrew/download_strategy.rb
+++ b/Library/Homebrew/download_strategy.rb
@@ -473,7 +473,7 @@ class CurlDownloadStrategy < AbstractFileDownloadStrategy
       url = url.sub(%r{^https?://#{GitHubPackages::URL_DOMAIN}/}o, "#{domain.chomp("/")}/")
     end
 
-    parsed_output = curl_head(url.to_s, timeout: timeout)
+    parsed_output = curl_head(url.to_s, wanted_headers: ["content-disposition"], timeout: timeout)
     parsed_headers = parsed_output.fetch(:responses).map { |r| r.fetch(:headers) }
 
     final_url = curl_response_follow_redirections(parsed_output.fetch(:responses), url)
diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index 65431ccf40..8ac14dd951 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -10,6 +10,8 @@ module Homebrew
     #
     # @api private
     module Strategy
+      extend Utils::Curl
+
       module_function
 
       # {Strategy} priorities informally range from 1 to 10, where 10 is the
@@ -53,14 +55,6 @@ module Homebrew
         "--silent"
       ].freeze
 
-      # `curl` arguments used in `Strategy#page_headers` method.
-      PAGE_HEADERS_CURL_ARGS = ([
-        # We only need the response head (not the body)
-        "--head",
-        # Some servers may not allow a HEAD request, so we use GET
-        "--request", "GET"
-      ] + DEFAULT_CURL_ARGS).freeze
-
       # `curl` arguments used in `Strategy#page_content` method.
       PAGE_CONTENT_CURL_ARGS = ([
         "--compressed",
@@ -188,11 +182,12 @@ module Homebrew
         headers = []
 
         [:default, :browser].each do |user_agent|
-          output, _, status = curl_with_workarounds(
-            *PAGE_HEADERS_CURL_ARGS, url,
-            **DEFAULT_CURL_OPTIONS,
+          output, _, status = curl_head(
+            url,
+            wanted_headers:    ["location", "content-disposition"],
             use_homebrew_curl: homebrew_curl,
-            user_agent:        user_agent
+            user_agent:        user_agent,
+            **DEFAULT_CURL_OPTIONS,
           )
           next unless status.success?
 
@@ -216,7 +211,7 @@ module Homebrew
       def self.page_content(url, homebrew_curl: false)
         stderr = T.let(nil, T.nilable(String))
         [:default, :browser].each do |user_agent|
-          stdout, stderr, status = curl_with_workarounds(
+          stdout, stderr, status = curl_output(
             *PAGE_CONTENT_CURL_ARGS, url,
             **DEFAULT_CURL_OPTIONS,
             use_homebrew_curl: homebrew_curl,
diff --git a/Library/Homebrew/utils/curl.rb b/Library/Homebrew/utils/curl.rb
index b32da61da0..4a6ca0cedd 100644
--- a/Library/Homebrew/utils/curl.rb
+++ b/Library/Homebrew/utils/curl.rb
@@ -205,7 +205,7 @@ module Utils
       curl_with_workarounds(*args, print_stderr: false, show_output: true, **options)
     end
 
-    def curl_head(*args, **options)
+    def curl_head(*args, wanted_headers: [], **options)
       [[], ["--request", "GET"]].each do |request_args|
         result = curl_output(
           "--fail", "--location", "--silent", "--head", *request_args, *args,
@@ -216,9 +216,9 @@ module Utils
         if result.success? || result.exit_status == 22
           parsed_output = parse_curl_output(result.stdout)
 
-          # If we didn't get a `Content-Disposition` header yet, retry using `GET`.
-          next if request_args.empty? &&
-                  parsed_output.fetch(:responses).none? { |r| r.fetch(:headers).key?("content-disposition") }
+          # If we didn't get any wanted header yet, retry using `GET`.
+          next if request_args.empty? && wanted_headers.any? &&
+                  parsed_output.fetch(:responses).none? { |r| (r.fetch(:headers).keys & wanted_headers).any? }
 
           return parsed_output if result.success?
         end