diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index 3f98a9faf4..baa2eea270 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -35,13 +35,24 @@ module Homebrew
       # `Utils::Curl` method calls in {Strategy}.
       CURL_PROCESS_TIMEOUT = CURL_MAX_TIME + 5
 
+      # The maximum number of redirections that `curl` should allow.
+      MAX_REDIRECTIONS = 5
+
+      # This value is passed to `#parse_curl_output` to ensure that the limit
+      # for the number of responses it will parse corresponds to the maximum
+      # number of responses in this context. The `+ 1` here accounts for the
+      # situation where there are exactly `MAX_REDIRECTIONS` number of
+      # redirections, followed by a final `200 OK` response.
+      MAX_PARSE_ITERATIONS = MAX_REDIRECTIONS + 1
+
       # Baseline `curl` arguments used in {Strategy} methods.
       DEFAULT_CURL_ARGS = [
         # Follow redirections to handle mirrors, relocations, etc.
         "--location",
+        "--max-redirs", MAX_REDIRECTIONS.to_s,
         # Avoid progress bar text, so we can reliably identify `curl` error
         # messages in output
-        "--silent",
+        "--silent"
       ].freeze
 
       # `curl` arguments used in `Strategy#page_headers` method.
@@ -183,7 +194,7 @@ module Homebrew
           )
           next unless status.success?
 
-          parsed_output = parse_curl_output(output)
+          parsed_output = parse_curl_output(output, max_iterations: MAX_PARSE_ITERATIONS)
           parsed_output[:responses].each { |response| headers << response[:headers] }
           break if headers.present?
         end
@@ -217,7 +228,7 @@ module Homebrew
 
           # Separate the head(s)/body and identify the final URL (after any
           # redirections)
-          parsed_output = parse_curl_output(output)
+          parsed_output = parse_curl_output(output, max_iterations: MAX_PARSE_ITERATIONS)
           final_url = curl_response_last_location(parsed_output[:responses], absolutize: true, base_url: url)
 
           data = { content: parsed_output[:body] }
diff --git a/Library/Homebrew/utils/curl.rb b/Library/Homebrew/utils/curl.rb
index f1dd65e450..d9687d670a 100644
--- a/Library/Homebrew/utils/curl.rb
+++ b/Library/Homebrew/utils/curl.rb
@@ -393,13 +393,17 @@ module Utils
     # `:status_code`, `:status_text`, and `:headers`.
     # @param output [String] The output text from `curl` containing HTTP
     #   responses, body content, or both.
+    # @param max_iterations [Integer] The maximum number of iterations for the
+    #   `while` loop that parses HTTP response text. This should correspond to
+    #   the maximum number of requests in the output. If `curl`'s `--max-redirs`
+    #   option is used, `max_iterations` should be `max-redirs + 1`, to
+    #   account for any final response after the redirections.
     # @return [Hash] A hash containing an array of response hashes and the body
     #   content, if found.
-    sig { params(output: String).returns(T::Hash[Symbol, T.untyped]) }
-    def parse_curl_output(output)
+    sig { params(output: String, max_iterations: Integer).returns(T::Hash[Symbol, T.untyped]) }
+    def parse_curl_output(output, max_iterations: 5)
       responses = []
 
-      max_iterations = 5
       iterations = 0
       output = output.lstrip
       while output.match?(%r{\AHTTP/[\d.]+ \d+}) && output.include?(HTTP_RESPONSE_BODY_SEPARATOR)