Step 1: 验证当前系统OpenSSL版本，并检查yum中目前有效的版本

openssl version

yum info openssl

Installed Packages
Name : openssl
Arch : x86_64
Version : 1.0.1e
Release : 57.el6
Size : 4.1 M
Repo : installed
From repo : base
Summary : A general purpose cryptography library with TLS implementation
URL : http://www.openssl.org/
License : OpenSSL
Description : The OpenSSL toolkit provides support for secure communications between
 : machines. OpenSSL includes a certificate management tool and shared
 : libraries which provide various cryptographic algorithms and
 : protocols.

Available Packages
Name : openssl
Arch : i686
Version : 1.0.1e
Release : 57.el6
Size : 1.5 M
Repo : base
Summary : A general purpose cryptography library with TLS implementation
URL : http://www.openssl.org/
License : OpenSSL
Description : The OpenSSL toolkit provides support for secure communications between
 : machines. OpenSSL includes a certificate management tool and shared
 : libraries which provide various cryptographic algorithms and
 : protocols.

Step 2: 下载最新版本的OpenSSL，目前1.0.2最新版本是1.0.2u

cd /usr/local/src
wget https://www.openssl.org/source/openssl-1.0.2u.tar.gz
tar -xvzf openssl-1.0.2u.tar.gz

Step 3: 如何手动编译和安装OpenSSL

cd openssl-1.0.2u
./config
make depend
make
make test
make install

Step 4: 移动最新安装的OpenSSL文件到系统路径下

mv /usr/bin/openssl /root/
ln -s /usr/local/ssl/bin/openssl /usr/bin/openssl

Step 5: 验证最新安装OpenSSL版本

openssl version
OpenSSL 1.0.2u 20 Dec 2019

资料来源：
https://techglimpse.com/upgrade-openssl-linux-tutorial/
https://webhostinggeeks.com/howto/how-to-install-and-update-openssl-on-centos-6-centos-7/